How would P2P Kriegspiel be designed?
It would use secure 2-party computation. There would probably be commitments to the statuses of the pieces and oblivious transfers to simulate referee responses and zero-knowledge protocols to show that the players are behaving as they should with respect to the above 2 parts.

If you're taking the oblivious transfer functionality as given and assuming the players are semi-honest, then I'm willing to try coming up with more details here. ​ On the other hand, if you're interested in how the oblivious transfers would be performed or how the zero-knowledge protocols would work, then you should ask on crypto.