I am told that, under FAA rules, a ppl accompanied by an instructor can log P1 if he holds a licence for the class of airplane (sic) and is occupying the left hand seat. We would call that P/UT, unless the instructor is acting as a passenger and not in command of, nor logging, the flight.
If the instructor is in the RHS to go for a jolly and get his hours up under the FAA system and does not in anyway instruct you then you would log P1 under JAA and he would log whatever they do in the US. I did 100 hours out in the US and became Mr Popularity as FAA instructors with time on their hands would want to come for a ride and enjoy a nice sleep whilst logging hours. Nice work if you can get it!