Every time I've purchased a used car from a real person, getting it to my mechanic for a once-over has never been a problem. Usually, they've let me take the car there myself pre-purchase, sometimes wanting me to leave some sort of collateral (such as the car I drove to them in). They already know where I live and what my phone number is, so if I steal their car, the cops won't have any problem finding me.

A couple of times, I've been fine with buying the car first, but with a clause in a sales contract saying that I have the right to return the car within 3 days for a full refund (basically replicating the effects of my state's lemon law. The lemon law only applies to new car purchases).

But the mechanic's check has never actually revealed any issues that the seller hadn't already disclosed to me. People tend to be pretty honest about those sorts of things.