There is a lot of confusion because physicists changed the meaning of “locality” since the EPR paper to refer to relativistic locality (sending information faster than light) which was not what Einstein was on about. Einstein’s locality is probably most succently summarized as such:

• ∀x(Var(Pr(S’|S))=Var(Pr(S’|S∪x))) where x∉S

In this case, assume a bunch of particles are interacting, and S is the state of a system of interacting particles prior to the interaction, and S’ is the state of the system of interacting particles after the interaction. We then want to look at the variance (statistical spread) of the probability distribution of S’ preconditioned on S, that is to say, a prediction of the state of the system after the interaction given complete knowledge of the state of the system prior to interaction, and then compare that to the variance of another prediction where we precondition both on S and x, where x is the state of something outside of the system of interacting particles.

If a theory is local, then the two should always be equal for any possible value of x. This is because the outcome of a local interaction should only be determined by everything participating in the local interaction, that is to say, S, so preconditioning on complete knowledge of the initial states of everything participating in the interaction should give you sufficient knowledge to predict the outcome of the interaction, that is to say, S’, to best that is physically possible.

If you can include something outside of the interaction, that is to say, x, and it can improve your prediction further, then it must be nonlocal because it contains irreducible dependence upon something not involved in the interaction.

The point about the EPR paper is that if you don’t assume hidden variables, then this definition of locality is broken. Two entangled particles are said to be ontologically in a superposition of states, meaning, having complete knowledge on their states prior to the measurement interaction can only predict them both with a distribution of 50%/50%, but if you precondition on knowledge of an observer’s measurement far away, then you can improve your prediction as to your measurement of your local particle to 100% certainty, which violates this locality condition.

This is still local in the classical case where the only reason you could improve your prediction is because you were ignorant of the initial state of the particle to begin with, so you never preconditioned on the complete initial state of the system to begin with. Hence, adding hidden variables would, supposedly, restore this notion of locality, which we can call causal locality as opposed to relativistic locality.

What Bell’s theorem proves is that adding hidden variables does not restore causal locality. This is because, as he proves, in quantum mechanics, the state of an individual particle in a collection of entangled particles can have dependence upon the configuration of a collection of measurement devices, even though it only ever interacts with an individual measurement device. That means this violation of causal locality is intrinsic to the mathematics of the theory and is not something that just arises due to a lack of hidden variables.

Even worse, as Bell says, adding hidden variables appears to make it “grossly nonlocal,” which by that he meant it violates relativistic locality as well. At least without introducing something like superdeterminism or retrocausality.

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