Such a sale is perfectly routine (some studies estimate that 40 per cent of US gun sales are of this type), requires no background check or records, and of course owning such a weapon is perfectly legal for an American.
Ah yes the study often trotted out as the 40% of purchases are done through the "gun show loophole". The study, by the way, was conducted in 1994 so it's almost old enough to drink and covered the acquisition period of 1993 and 1994 and collected data on guns owned prior to 1993, likely because information on purchases made a decade prior would be sketchy at best. In case anyone is interested the Brady Act which instituted federal background checks was passed at the end of November 1993 and became effective February 1994.
Oh and about that loophole, only 4% said they bought it at a gun show or flea market. Don't believe it, read the pdf here and see for yourself. Spoiler alert, it's Exhibit 5 on page 6. It shows that 60% of acquisitions were through a store of some sort, hence the 40%. Funny how nobody mentions the 24% that were either gifts or inherited.