The only person who is able to log P1 time is the Captain which is the person who signs the technical log. In a single pilot aircraft the captain must occupy the primary seat; i.e. the left seat, unless for the the purposes of instruction as part of a recognised course of training or testing.
There have been legal cases, which I am not prepared to quote here because of professional confidentiality, where pilots have claimed experience as P1 but on subsequent post incident investigation it has been found that they did not sign the technical log on the flights claimed as P1.
If one of the pilots is the only insured person he must be the captain, must sign the technical log, must occupy the primary seat and is the only person allowed to claim P1.
Certified multi-pilot rules are different but for single pilot aircraft it is my opinion that the above applies.
MM