I've answered one of these calls and it turned out to be a 'nothing job' as we say in the ambulance service. I wasn't asked to provide credentials at that stage; I guess the way the conversation went following the announcement was enough. I left full details of both myself and my then employer, then was very well looked after for the rest of the flight and received a thank you letter to boot. The captain asking me if we needed to divert tempted the mischievous side in me but I remained sensible
As for the other call on a different flight, my mate (Paramedic) and I shrunk into out seats as we'd had a beer and left it to someone else.