Fireflybob,
It is not a 'joint responsibility' to maintain licence validity, this responsibility lies with the licence holder alone. Any decent company however will have a system in place to remind the individual that a certain element of his licence is due for renewal as it is obviously in their interest to make sure that the employee remains employable at all times. Nonetheless if you hold a licence it is up to you and you alone to ensure that it remains valid.
Smudger, yes from a legal standpoint I am aware of this and I agree with you but I believe there is also an onus on the operator to ensure their crews are operating legally.
Slightly off thread but several years ago a couple of pilots for a UK airline exceeded 100 hours in 28 days. Both the pilots AND the operator were (successfully) taken to court by the Authority.
If a pilot is flying with an invalid licence I still think the operator has some accountability.