[El Gato]

All private dealers conduct background checks.

There is no such thing as a private dealer. If you are a dealer, you must have a FFL, and conduct the NICS check, at your place of business, or at a gun show. (With some exceptions, such as purchases by those with CHL/CCW from certain states, like Texas for example).

That said, collectors and/or hobbyists can sell guns, at a gun show, or elsewhere without conducting any check at all, at least in the Free States. YMMV. (It would be good to check a driver's license, since even for individuals, sales to out of state residents may be illegal.)

[tet68]

Collectors under an O3 FFL can sell to individuals but are
still required to note the details of the disposition in their bound book, identification,DL# etc, make a mistake and you could be in deep trouble.
The ATF can demand at any time , an inspection of your collection or a review of your bound book.

[Darren McCarty]

All dealers are FFL's, and by private dealer, I'm referring to dealers not owned by the government.

I wasn't referring to individuals in private sales which are not dealers.