[m_the0ry]

This is definitely an interesting hypothetical, but I had a little trouble trying to get a meaningful answer.

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In one half of the ship, the thermodynamic arrow of time points in one direction, and in the other half of the ship, it points in the opposite direction.

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I realize this is a thought experiment, but we call the second law a 'law' for a reason. As in, this is a rule that must be symmetric and invariant for the entire universe. Therefore I think you inherently must be describing two different universes, because their laws are different. How can two universes with two different sets of laws interact? That's a damn good question that no one will claim to have the answer to. I personally think, however, that no meaninful information exchange can take place between two universes because the asymmetry of their 'laws' _must_ destroy the context of all information.

Now, it's possible that all the processes on both sides are isentropic (adiabatic with respect to the 'barrier' and fully reversible). If this is the case, both sides would see the other side in a state of rest. In other words there would be no apparent 'time evolution'.