If it is not a training flight you can claim P1 and the instructor claims nothing. The best bet is to agree before the flight who is logging the hours. The key thing is you cannot both claim P1 and if you claim P2 it will have to be as PUT or PICUS.
If it is not a training flight then what on earth would the instructor be doing there anyway? I can't believe any instructor would go flying and not log it. What's he there for?