I've purchased two firearms from private owners exactly as you described. Here in Ohio, both being residents of the same state (I haven't a clue if it's legal in TN) we wrote out a receipt/bill of sale. Check a drivers license for ID verification (if they refuse, abort the sale) and write down name, address, drivers license number, date of sale, type of gun sold/bought, the serial number, etc. Both sign it and you save your copy as long as you own the gun. If ATF knocks on your door wanting info on that gun you just bought/sold, you have a signed receipt. It's CYA! You really need to research gun selling/private owner purchases laws of TN.
No one reads or cares what is written in ones signature box. So I'm not writing anything worth reading or remembering.