For much of America, this is a false statement. At least a third of all states have passed background check requirements on private sales and many others require both the buyer and seller to have valid ID in their state. The reason that they do this is because both parties in private sales have to reside in the same state or else the seller needs a FFL. You cannot prove residency without seeing a valid ID, so they require the ID to be presented.
Using Florida as an example, in order to be legal, private sales must meet 3 conditions: the buyer and seller must both possess a valid ID issued by the state of Florida, the buyer must meet the age requirements, and the buyer must not be a prohibited person.
https://www.tampacarry.com/florida-rules-on-private-firearm-sales
Similar to voter regulations, while private firearm sales are more wide-open in federal law as long as both parties reside in the same state and the transaction is within this state, the states are free to regulate the sales and almost all of the do so. The Constitution is at play here with federal government's inability to regulate intrastate commerce even though they can regulate interstate commerce.