You can purchase firearms for cash from many dealers (at gun shows) without either a background check or a paper trail.
However, all new guns are (since 1968) purchased from a licensed dealer, so there is always the beginning of a paper trail. How far that goes before it gets muddled in private sales is another matter.
My understanding is that there is technically an exception to this where you can manufacture your own firearms so long as you stay under the yearly limit and do not manufacture them with the intent of distributing them. Once you have done so, you can then distribute them as a private citizen.
That reads weird because the legality of distributing it depends on if you can assert that distribution was not your intent when you made it. This means this is probably not a very practical exception, as it potentially involves you arguing something rather nebulous in court.
If they are a private individual (not in the business of selling firearms) then yes. But that's no different than buying a used deer rifle from your neighbor.