Interesting debate folks.
I regularly fly with a CFI from a local flying club ouside his official capacity. Before taxi, it is always discussed who will be responsible for the flight ie EFATO. If I am the Captain for the trip I deal with the flying and landing (safely hopefully) and he will deal with radios etc if required and vice versa. But each of us know who is doing what. In the air two heads and eyes are better than one and a team discussion (time permitting)can be had for the safe outcome of the trip if necessary. I also fly with a multi flight deck crew where everyone has an imput and the Captain has the final say. If one person doesn't like the conditions (wx for example) then it doesn't happen. CRM. Final decision still rests with the Captain. Pressonitis and bravado are a recipe for disaster.
I always understood that if you were P1 for a portion of the flight you could log that and P2 the rest.