I'm pretty sure we hoi polloi don't know for sure, but I thought the best guess in general is that the operating carrier of the qualifying flight pays for lounge access, since lounge access is generally a perk to encourage lounge-eligible customers to travel on a given airline (even though US airlines generally view the revenue from a lounge membership as more valuable than the marketing incentive). So if you're flying AS and entering the AS lounge, I'm not sure any money would exchange hands (except AS is losing the opportunity to charge you for a lounge visit either through a membership or some other paid access), whereas if you're flying AS and using the BA lounge, AS would pay BA (and vice versa) so scanning the ticket is more important. But again, I certainly don't claim to know with any certainty how the agreements are written.

This would clearly make sense for long haul flights: if you're flying QF F and using a BA F lounge, clearly QF would pay BA. If you're flying QF Y as an AS Emerald using a BA lounge, seems to make sense, since it's QF you're paying with airfare.

Whatever the underlying reason, the rules are clear even if you do typically get away with trying to double-dip on benefits from different programs in this way. No matter who's supposed to be paying whom, it's probably difficult or impossible if your FF# isn't in the record.