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.