Your hypothetical has nothing to do with quantum mechanics or many worlds, and everything to do with special relativity.
Special relativity hangs out in a nice, flat, well behaved, Minkowski space where this sort of thing cannot happen. It takes general relativity and specifically a universe with accelerating expansion (such as ours probably is).
It can also happen in flat space, e.g. if you and the other agent are on Rindler trajectories accelerating in opposite directions, then nothing that one of you does can affect the other.
Special relativity hangs out in a nice, flat, well behaved, Minkowski space where this sort of thing cannot happen. It takes general relativity and specifically a universe with accelerating expansion (such as ours probably is).
It can also happen in flat space, e.g. if you and the other agent are on Rindler trajectories accelerating in opposite directions, then nothing that one of you does can affect the other.