It's widely held that it's either very difficult or impossible to affect the outcomes of experiments in other branches of the many-worlds.
The reason given is usually something along the lines that the "worlds" are orthogonal, and each world obeys the Schrödinger equation on its own as if the other worlds didn't exist.
But there is still a state vector for the whole system in a higher dimensional Hilbert space. We don't need to worry about the whole universe here-- we can start with a single quantum coin flip in a sealed laboratory and take it from there.
The state vector evolves and its evolution can be influenced-- one of the mes in this lab after the first coin flip can switch magnets on and off etc, do what he likes. He could even set off a hydrogen bomb in the middle of the lab.
What stops him making changes that the other "him" can perceive? It seems that whatever he does he can only affect the lab's relative state relative to him, and not the other guy's. He can cause the state vector to move, but only in a subspace the other him is unaware of.
The MWI is usually expressed in terms of information and observations. But it seems like the same rules must apply in the other direction-- i.e. causality has to be restricted as well. Somehow causality, as well as observation, is also a kind of entanglement.
(Later update) OK I think I might have figured this out.
Our system starts in the state |lab>. After I flip the coin, it evolves into a |lab>|H> + b |lab>|T> where |x>|y> is the tensor product of |x> and |y> and a and b are complex coefficients, each with the value 1/sqrt(2) in this case.
Now, any action that the me who got heads now performs must be observable by me. All observations must obey the normal probability rules. This means that however the left-hand component of that expression evolves, its overall coefficient of 'a' in the combined system cannot change.
Suppose I flip another coin, for example. The state evolves to:
a (c |lab>|H>|H> + d |lab>|H>|T>) + b |lab>|T>
c and d have to be normalized on their own, or my second coin flip would not be empirically faithful. The whole expression (representing the state of the branched universe) also has to remain normalized. It follows from this that c and d can only divide up the contribution made to the total by a, and cannot affect anything else in the whole expression. Unless there are interference effects. If there are, then I can alter things in the a * b and a b * terms which would be observable by the other me.
The fact that the initial coin flip states |H> and |T> are orthogonal kills any interference terms and is therefore said to "decohere" the system into two non-interfering components. But this leaves room in theory for communication between the worlds so long as decoherence can be prevented (which is what the Plaga experiment for example is proposing).
Or so people say. It seems to me that the mathematical description of Alice communicating with herself across universes is mathematically identical to EPR-like setups where Alice tries to communicate with Bob remotely with an entangled pair of particles. In the latter case we have two correlated coins in a single universe, and in the former case two correlated coins in two universes.
The more correlated Alice's coin is with Bob's, the less she knows about her coin on its own. So she can only send Bob random "messages". The same exact problem applies to Alice trying to talk to the other Alice.