> Sure, but the momentum doesn't determine the position (due to the constant of integration) so you can have two particles with the same momentum functions and different locations, and that leads to my next question...

There's no constant of integration since the integral will be over all of space (or momentum space).

> That's what I actually meant to ask but didn't phrase clearly: since position and momentum are described by wave functions, can you entangle the positions of two particles? or entangle their momentum?

Certainly! I couldn't think of any examples of how it might occur, but I googled it and found an answer on Quora that seems to be correct: https://www.quora.com/In-quantum-mechanics-how-do-I-comprehe...