While Tennessee mandates federal background checks by licensed firearm dealers who must retain records pursuant to federal law, the Tennessee bureau of investigation must destroy all such records associating a particular individual with a particular firearm purchase. Background checks are not required for sales between private individuals, however, that is not to say private sellers should not keep record of such sales.
It is a state crime for any person to purchase or attempt to purchase a firearm knowing they are legally prohibited from doing so, and likewise is a crime for anyone to sell a firearm to a person knowing that person is legally prohibited from owning a firearm.