I recently sold a car, and I wasn't in possession of the log book for it. I'd sent the green slip away months ago, and never had the log book back. I assumed I had so advertised it.
Wifey reminds me, so I went to local DVLA office to apply for the book (cost £25) and got a receipt for the same. Bloke came to my house, took the car away and a few weeks later the log book came, so I signed it and forwarded it to the new owner.
I suppose it's all a matter of trust.