This post is a restatement of a Physics Forum post answering the question "is gravity action at a distance or is it a local force" and a specific subquestion related to string theory and loop quantum gravity which I include in my answer:
Gravitational Waves and Hypothetical Gravitons Propagate At The Speed Of Light
The affirmative detection of gravitational waves by LIGO and other gravitational wave detectors, coinciding to within error bars of photon evidence of a black hole-neutron star merger, strongly supports the view that gravity is a local effect that propagates at the speed of light, rather than an instantaneous "at a distance" effect.
General relativity, and every graviton based quantum gravity theory adopts this position.
Localization Issue In Classical General Relativity And The Issues It Poses For Quantum Gravity
This said, however, there are issues like the localization of gravitational energy, and the self-interactions arising from gravitational fields, which conventional general relativity theory as conventionally applied does not recognize.
Similarly, while individual gravitational interactions in general relativity conserve mass-energy, in general relativity with a cosmological constant the aggregate amount of mass-energy in the universe (apart from gravitational potential energy) increases with time at a global level, although it is arguably possible to describe a gravitational potential energy concept to address this which IMHO doesn't really work but a minority of GR theorists say overcomes the mass-energy conservation issue.
The theoretical aspects of GR that disfavor localization or don't conserve mass-energy area particularly problematic when trying to quantize gravity, because a quantum particle based theory pretty much entirely has to be a bottom up theory that derives all global properties from individual local interactions that the theory permits.
Entanglement and Locality
In quantum gravity theories, it is conceivable that gravitons could become entangled with each other leading to correlations in the properties of the entangled particles analogous to those seen in quantum mechanics in other contexts.
But entanglement effects always involve particles that share a light-cone in space-time, and as a practical matter, while we can measure the correlated properties of entangled photons or other Standard Model particles, we do not have the capacity to measure the properties of individual gravitons, either individually, or statistically, so there is no way to resolve entanglement questions related to quantum gravity in the straightforward direct way that we do with Standard Model particles.
The gist of entanglement is that the correlations observed require the sacrifice of at least one of three axioms that we can usually resort to in physics: causality, locality, or realism. So, while sacrificing locality is one way to get entanglement-like effects, it is not the only one. Arguably, they are all equivalent ways of expressing the same concept and the fact that this is not manifestly obvious simply indicates that our language is not aligned with underlying concepts of how Nature really works.
In the case of interactions involving photons, gravitons and gluons, I personally often find it convenient to, and tend to favor, sacrificing "causality" (i.e. the arrow of time) rather than locality, because fundamental massless spin-1 bosons always move at the speed of light, and hence, do not experience the passage of time in their own reference frame, so it makes sense that interactions mediated by fundamental massless spin-1 bosons should not experience an arrow of time either, thus disallowing CP violation (which empirically is not observed) in interactions mediated by these massless bosons, and essentially stating that the line in space-time that entangled massless bosons follow basically amount to simultaneous points in the space-time coordinates that are best suited to judging causality. But to some extent the decision regarding which axiom to sacrifice in entanglement situations is a stylistic one with no measurable real world effects.
But, lots of loop quantum gravity oriented quantum gravity theories adopt causality as a bedrock axiom and treat the dimension of time in some sense distinctly for space dimensions, so one must sacrifice either locality or reality to some degree in these causation affirming LQG theories.
Similarly, setting up a graviton entanglement experiment (or even a "natural experiment" that would entangle gravitons somehow so that we could measure these effects) is beyond our practical experimental and observational capacity.
Another possible angle to get at this issue which is attracting attention is to look at a phenomena in which a group of Standard Model particles acts "coherently" in the absence of outside interactions. In ordinary daily life, we are bombarded by also sorts of particles that leads to rapid decoherence except in rarified circumstances. But in a deep space vacuum, a coherent group of particles can be expected to travel vast distances with only slight non-gravitational interactions with the outside environment.
Theorists can use even very incomplete quantum gravity theories in which lots of quantities can't be calculated to then calculate the extent to which a flux of gravitons would lead to decoherence of that group of Standard Model particles sooner than it would in the absence of such interactions (see, e.g., here).
The rate at which decoherence emerges in objects in the deep vacuum is thus a physical observable that helps us tease out the mechanism by which gravity works.
Non-Local Gravity Theories
There are explicitly non-local formulations of gravity and papers on this topic address a lot of the issues that the OP question seems to be getting at. Rather than try to explain them myself, I'll defer to the articles from the literature below that discuss these theories.
Literature For Further Reading
Some recent relevant papers include:
* Ivan Kolář, Tomáš Málek, Anupam Mazumdar, "Exact solutions of non-local gravity in class of almost universal spacetimes" arXiv: 2103.08555
* Reza Pirmoradian, Mohammad Reza Tanhayi, "Non-local Probes of Entanglement in the Scale Invariant Gravity" arXiv: 2103.02998
* J. R. Nascimento, A. Yu. Petrov, P. J. Porfírio, "On the causality properties in non-local gravity theories" arXiv: 2102.01600
* Salvatore Capozziello, Maurizio Capriolo, Shin'ichi Nojiri, "Considerations on gravitational waves in higher-order local and non-local gravity" arXiv: 2009.12777
* Jens Boos, "Effects of Non-locality in Gravity and Quantum Theory" arXiv: 2009.10856
* Jens Boos, Jose Pinedo Soto, Valeri P. Frolov, "Ultrarelativistic spinning objects in non-local ghost-free gravity" arXiv: 2004.07420
The work of Erik Verlinde, for example, here, also deserves special mention. He has hypothesized that gravity may not actually be a distinct fundamental force, and may instead, be an emergent interactions that arises from the thermodynamic laws applicable to entropy and/or entanglement between particles arising from Standard Model interactions.
His theories approximate the observed laws of gravity, sometimes including reproduction of dark matter and/or dark energy-like effects, although some early simple attempts that he made to realize this concept have been found to be inconsistent with observational evidence.
Does string theory or loop quantum gravity have hypotheses on what gravity is or how it arises?
In string theory, either a closed or open string gives rise in certain vibration patterns to gravitons which carries the gravitational force between particles in a manner rather analogous to photons that utilizes a "loophole" in key "no go theorems" related to quantum gravity that make a naive point particle analogy to photons not viable.
This is generally done in a 10-11 dimensional space, although the way that the deeper 10-11 dimensions are distilled to the three dimensions of space and one dimension of time that we observe varies quite a bit. In many versions, the Standard Model forces as manifested through string theory are confined to a four dimensional manifold or "brane" while gravitons and gravity can propagate in all of the dimensions.
The distinction between the dimensions in which the Standard Model forces can operate and those in which gravity can operate helps string theorists explain why gravity is so weak relative to other forces, relative to the generic naive expectation of versions of string theory that if all forces ultimately derive from universal string-like particles, they ought to be more similar in strength, especially at high energies.
There are multiple problems with string theory but the biggest one is that it is really a class of vast numbers of possible theories that do not uniquely give rise to a single low energy approximation that resembles the Standard Model, and nobody has figure out how to thin out the universe of possible low energy approximations of string theory to find even one that contains everything that the Standard Model contains, while lacking everything that we have no experimental evidence for at experimentally testable energies. So basically, there are almost no observables that can be calculated from string theory.
String theory, for example, tends to favor (and arguably requires) that its low energy approximations be supergravity theories (a class of theories that integrates supersymmetry theories with supergravity theories), Majorana neutrinos that undergo neutrinoless double beta decay, proton decay, and models in which the initial state of the Universe at the Big Bang has baryon number zero, lepton number zero, and engaged in baryogenesis and leptongenesis soon after the Big Bang with a high energy process showing CP violation, baryon number violation and lepton number violation, that generates far more particles than the only known Standard Model processes that do so. The existence of a massless spin-2 graviton is pretty much the only prediction of string theory that has any support from observational evidence, and of course, that itself, is only indirect and in its infancy. But the mathematical intractability of quantum gravity by other means under various "no go theorems" has been one important motivation for string theory's popularity.
In loop quantum gravity, the universe is fundamentally made up of nodes that have a small finite numbers of connections to other nodes, and gravity is quantized primarily by quantizing space-time, rather than primarily by quantizing particles against a background that is smooth, continuous and local.
In LQG, locality is ill defined at this most fundamental level and is only an emergent property of the collective interactions of all of the nodes in a sense similar to that of temperature and pressure in the thermodynamics of gases being emergent properties of individual gas atoms randomly moving around a particular speeds that can be described globally in a statistical fashion. Particles move from node to node according to simple rules.
For example, LQG imagines that in space-time, most nodes in what we perceive to be a local area of space-time will connect to other, basically adjacent, nodes in the same local area, but there is no fundamental prohibition on a node having some connections to nodes in what we perceive to be the same local area, and other connections to nodes in what we perceive to be a local area billions of light years away.
The number of space-time dimensions is likewise an emergent property in LQG, and the number of space-time dimensions that emerge in this fashion aren't necessarily integer quantities. A system of nodes could also be described with a fractal dimension that is not an integer defined in a manner similar or identical to the mathematical definition of a fractal dimension.
Some edge examples of LQG theories think of matter and particles themselves as deformations of space-time itself that are emergent, rather than as something separate that is placed within a space called "space-time."
As in classical general relativity, gravity is fundamentally a function of the geometry of space-time, but in LQG, that geometry is discrete and broken rather than smooth and continuous, and locality is (as discussed above) ill defined. In LQG, the "background independence" of the theory, realized by not having a space-time distinct from gravity, is a hallmark axiom of the field and line of reasoning involved. This has the nice feature of "automatically" and obviously giving LQG properties like co-variance that impose tight constraints on the universe of gravity theories formulated with more conventional equations of gravity, like the Einstein field equations, which have this property, even though this is not obvious without extended and non-obvious mathematical proofs. But it has the downside of expressing how gravity works in equations that are not very conceptually natural to the uninitiated, which can make understanding what LQG really says in more familiar contexts challenging.
One of the biggest practical challenges for LQG when confronted with experimental evidence, is that many naive versions of it should give rise to slight Lorentz Invariance Violations (i.e. deviations from special relativity) at the Planck level due to the discrete rather than continuous nature of space-time, because Lorentz Invariance is formulated as a continuous space-time concept. Strong experimental constraints disfavor Lorentz Invariance Violations to levels that naively extend below Planck length scale distances. But, the problem of discrete formulations of space-time leading to minor deviations from Lorentz Invariance is not a universal property of all LQG theories and can be overcome with different conceptualizations of it that evade this problem.
Like string theory, LQG is very much a work in process that is striving to find ways within its general approach and family of theories that reproduce either classical general relativity in the classical limit, or a plausible modification of classical general relativity that can't be distinguished from general relativity with current observational evidence. There are a host of intermediate baby steps and confirmations of what is predicted that have to be surmounted before it can gain wide acceptance and produce a full spectrum of "big picture" results, in part, because so much of this class of theories is emergent, and has to be discovered, rather than being put in by hand as higher level operational and useful theories in practical situations.
Footnote Regarding Loop Quantum Gravity Terminology
There are two senses in which the term "loop quantum gravity" (LQG) is used, and I'm not being very careful about distinguishing the two in this post. Some of what I say about LQG is really specific only to the narrow sense theory that I discuss below, while other things that I say about LQG applies to the entire family of LQG style quantum gravity theories.
In a strict and narrow sense, loop quantum gravity refers to a specific quantum gravity theory that involves quantizing space-time that is largely attributed to Lee Smolin and Carlo Rovelli, although assigning credit to any scientific theory that a community of interacting researchers help formulate is a perilous and always somewhat controversial thing to do.
But the term is also frequently used as a catchall term for quantum gravity theories that share, with the narrow sense type example of loop quantum gravity, the feature that space-time itself or closely analogous concepts are quantized. LQG theories are distinguishable from quantum gravity theories, like string theory, that simply insert graviton particles that carry the gravitational force into a distinct pre-determined space-time with properties sufficient to be Lorentz invariant and observe other requirements of gravity theories that approximate general relativity in the classical limit.
For example, "causal dynamical triangulation" (CDT) is a quantum gravity theory that is in the loop quantum gravity family of theories, but is not precisely the same theory as the type example of LQG after which this family of theories is named.
"Spin foam" theories are another example of LQG family quantum gravity theories.