Verifying a club members pilot certificate and other paperwork

A quick question or four:
Does you club verify that a member has a license, a valid BFR, and/or
any other certification documents? Do you record the actual numbers,
photocopy, or just verify?
Is it different if that member is just towing with their private ship,
as opposed to flying a club glider?
Are there downsides to knowing or not knowing this information?
Specific, not hypothetical examples if possible. (yeah, I'm
dreaming...)

Thanks!
John