Scoobster,
A lot of flying regulation is carried out by insurance companies. When I started gliding many years ago, anyone could fly a glider in the UK provided they had third party insurance. You actually couldn't get this insurance unless you satisfied the sensible requirements laid down by the BGA.
Similarly, anyone with a current SEP rating can legally fly something like a C152 provided they have third party insurance. It is the conditions laid down by the insurance company which dictates what you can and cannot do. Every club I have ever belonged to insists that, to fly their aircraft, you have to be a club member and have signed the Flying Order Book.
For Group owned aircraft it is again the insurance which dictates what you can do. My group insurance has named pilots or instructors only but, if I change our insurance to, say, Visicover, they give the same cover for any pilot. In that case it would appear that you could allow another current SEP to log P1 time. I sometimes get asked to take up aspiring airline pilots who are desperately trying to keep current while they look for that first job. I have to point out that, while I am very happy to take them up, they cannot log any of the time as P1.