I've only ever bought one aeroplane. It wasn't terribly expensive (in absolute terms, it nearly broke me though
) and that was done exactly as you would do it for a car:
Person to person, private deal, no solicitors, no complications, just turn up with a load of dosh after an inspection (did it myself) and a test flight (didn't have a PPL at the time...)!
Now, how that would work on something a bit more valuable, I don't know!