There is no paperwork required to transfer a firearm privately, whether as a gift or as a sale, so long as both parties are Florida residents and are legally allowed to own a gun. In a private transfer, the legal age for a handgun is even lowered to 18 instead of 21.
Keep in mind, however, that it is a federal crime to knowingly transfer a firearm to anyone who cannot legally own one, even if it is just for a second. To be safe, it is a really good idea any time you are selling a firearm to another individual, you should make out a simple Bill of Sale in which youhave the transferee sign that they can legally purchase that firearm pursuant to federal and state law. It doesn't have to be complicated:
"I _________ am not legally, under state and federal law, banned from possessing a firearm"
This would take you two seconds to write out and have them sign andit wouldprotect youif it was later determined that that person was not allowed to own firearms.