Are you talking about both parties going to an FFL and transferring the weapon from one party to another?
There is no statute covering private sales in FL, except that the legal age of possession is 18. They are perfectly legal. Statutes, in general, tell you what you can't do, not what you can do. Find me a statute that sez it's legal to wear Wrangler jeans on a Tuesday.
About half the firearms I own came to me through private sales. I've only signed one bill of sale from one gentleman who asked that I do so.