have no experience with that other than flying in/on two different continents, but as far as I understood: if the plane is registered in a state that is a member or ICAO and the plane is of a category/type you are allowed to fly (type-rating and licence) you should be able to log your time as P1/P2 in your main log-book.
Actually, this brings to another question: I wonder if you can log time as a P2 on the same type but N-registered plane but have no FAA licence? I do not think so but am not sure.