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does the pilot wave theory respect locality?
EPR proves quantum mechanics violates locality without hidden variables, and Bell proves quantum mechanics violates locality with hidden variables, and so locality is not salvageable. People who claim quantum mechanics without hidden variables can be local tend to redefine locality to just be about superluminal signaling, but you can have nonlocal effects that cannot be used to signal. It is this broader definition of locality that is the concern of the EPR paper.
When Einstein wrote locality, he didn't mention anything about signaling, that was not in his head. He was thinking in more broad terms. We can summarize Einstein's definition of locality as follows:
(P1) Objects within set A interact such that their values are changed to become set A'. (P2) We form prediction P by predicting the values of A' while preconditioning on complete knowledge of A. (P3) We form prediction Q by predicting the values of A' while preconditioning on complete knowledge of A as well as object x where x⊄A. (D) A physical model is local if the variance of P equals the variance of Q.
Basically, what this definition says is that if particles interact and you want to predict the outcome of that interaction, complete knowledge of the initial values of the particles directly participating in the interaction should give you the best prediction possible to predict the outcome of the interaction, and no knowledge from anything outside the interaction should improve your prediction. If knowledge from some particle not participating in the interaction allows you to improve your prediction, then the outcome of the interaction has irreducible dependence upon something that did not locally participate in the interaction, which is of course nonlocal.
The EPR paper proves that, without hidden variables, you necessarily violate this definition of locality. I am not the only one to point this out. Local no-hidden variable models are impossible. Yes, this also applies to Many Worlds. There is no singular "Many Worlds" interpretation because no one agrees on how the branching should work, but it is not hard to prove that any possible answer to the question of how the branching should work must be nonlocal, or else it would fail to reproduce the predictions of quantum theory.
Pilot wave theory does not respect locality, but neither does orthodox quantum mechanics.
The fear of developing nonlocal hidden variable models also turn out to be unfounded. The main fear is that a nonlocal hidden variable model might lead to superluminal signaling, which would lead to a breakdown in the causal order, which would make the theory incompatible with special relativity, which would in turn make it unable to reproduce the predictions of quantum field theory.
It turns out, however, that none of these fears are well-founded. Pilot wave theory itself is proof that you can have a nonlocal hidden variable model without superluminal signaling. You do not end up with a breakdown in the causal order if you introduce a foliation in spacetime.
Technically, yes, this does mean it deviates from special relativity, but it turns out that this does not matter, because the only reason people care for special relativity is to reproduce the predictions of quantum field theory. Quantum field theory makes the same predictions in all reference frames, so you only need to match QFT's predictions for a single reference frame and choose that frame as your foliation, and then pilot wave theory can reproduce the predictions of QFT.
There is a good paper below that discusses this, how it is actually quite trivial to match QFT's predictions with pilot wave theory.
tldr: Quantum mechanics itself does not respect locality, hidden variables or not, and adding hidden variables does not introduce any problems with reproducing the predictions of quantum field theory.