Here's how I understand your question: 

A and B are space-like separated and make a measurement on a *single* particle that has equal (or just non-vanishing) probabilities of being in A's or B's region.

You now ponder how the measurement process works on a deeper level.
Could the collapse be a dynamical (i.e. time dependent) process? I think it can *not*. If it were, A and B would both try to 'pull' the wave-function to their side. You also get in trouble with relativity and the question who starts the process first.Thinking deeper, this will probably lead to the conclusion that the wave-function is not to be considered as 'real' (or 'ontic'). But I have not done this or seen this done.

It is thus better to regard the wave-function just as information. As soon as A or B have measured the particle, they have gained information and can update (=collapse) the wave-function.

**Back to the specific question**: For the physical outcome, it does not matter *WHEN* you collapse the wave-function. QM always assures that only one of A,B can measure the particle. Asking *'when'* the collapse 'occurs' is not a sensible question, because the collapse is not dynamic.

Added: If B knows the outcome of A, he must use the 'collapsed' wave function. If he does *not*, he must use what he knows, i.e. the original state X.