There are so many conditions on private sale that this answer is essentially incorrect. The vast majority of Americans live in a state that either requires that private sales go through an FFL (The federal firearm dealer license) or has other background check requirements with failure penalties on the seller severe enough that you'd want to protect yourself by going through an FFL anyway. And private sale across state lines is federally illegal, eg someone in Missouri can't do a private sale to someone from Oregon. And you cannot "speed up" a federal background check with money, the check is not run by the FFL themselves, they're just a trustworthy party that submits your paperwork to the feds. The wait time is entirely dependent on federal backlog and one's law enforcement records.
Law firm site with a list of private sale laws per state: https://www.findlaw.com/consumer/consumer-transactions/private-gun-sale-laws-by-state.html