Hello from Florida! I'm not a lawyer, and this is not legal advice, but here's one tolerably well read person's opinion.
Rules on background checks and such apply to FFL holders. One private citizen selling a personal weapon to another is a private matter. You need not ask for anything except proof that the person is over 21 (it's a crime here to sell a handgun to anyone under 21; for long guns, it's 18).
Note that under "local option", a private sale at a gun show may be subject to background check requirements. The Legislature here didn't want to p*ss off either the NRA or HCI so they ducked the decision and made it a county by county option (duh).
For the sake of prudence I would get a signed bill of sale with the buyer's name and address from the driver's license. It should state that the item is sold "as is" with no warranty, and that the buyer knows of no reason why he/she may not own the weapon. This is to cover your butt in case the cops come knocking, having found a gun at a crime scene and traced it to you through warranty records or such.
Of course, I would get a signed "as is" bill of sale for just about anything.
Check here for Florida gun laws info:
http://myweb.magicnet.net/~flynn/fla-gun.html
[This message has been edited by David Scott (edited June 15, 2000).]