My experience:
6 syndicates I've been in, 1 checked regularly and reminded members when they were due to expire, 5 who left it to the individual pilots.
Flying school I teach at part time has a whiteboard where instructors update their expiry dates themselves.
Flying schools where I've been a student / renter, about 50% check when you join, one asks for a confirmation of dates on the annual membership form, others check from time to time when checking legality of something (usually some test or other).
As a freelancer, nobody's ever asked to see my licence. I do make sure I've checked licence and logbook for pilots I've not flown before and am doing some training.
G