Probably not liable for letting someone drive your unfamiliar car. But I do agree with the consensus that unless you can verify that they have full comprehensive insurance covering your car (not just theirs) don't let them drive. Addresses, keys, other cars etc. aren't good enough. They might not own any of these things and there's no point suing someone with no money.
This is generally why I take the hit and trade in rather than sell privately.
My comprehensive cover with RAC does not allow me even to drive any other car at all, not even 3rd party!!!
They've heard about you, pal!