[recycled sixtie]

The last car I bought privately was my P car. No $$$ down and the seller let me take it to an indy garage for a PPI. Had a carfax done on it but showed no accident.
However the bundle of receipts for work done showed an accident by first owner incurring about $1100 to fix it so it was more of a fender bender.

The Miata I bought from a dealer showed me copies of work done on it. Privacy did not seem to be an issue. I got a ppi done before I bought it. It came up clean.
I cannot remember if I had to put $$$ down for a down payment.

If you find a car you like obviously have a ppi done but try and not put $$$ down as a down payment. Leave your own car there so they know you are coming back.
Ask for receipts for work done. Private sellers are more likely to hand over receipt info. That is why I like dealing privately plus they are not prof. negotiators. And don't fall in love with the first one you see!Enjoy your search.