All beaches in Florida are "public" from the tideline to the shore. That is state law, so if you access them from the water side and do not stray above the tideline, no one can keep you out.
As to Siesta, the land-side accesses at Siesta Beach (aka the North Beach) are all public, afaik. However, Crescent Beach in the center of Siesta Key is accessed almost entirely via private property. I think that you would be happier at Crescent, most likely. However, you should be aware that there are no food concessions or public toilets at Crescent; if you want snacks or drinks you'll have to carry them out there, and if you have to go potty you'll need to walk back to your room.
As to swimming in February -- no locals will deliberately be swimming in either of those places at that time of year. Falling in while surf-fishing perhaps, but not going in on purpose. AFAIC, no Florida beach north of Ft. Myers is warm enough for that in February. The average water temp in Sarasota County in February is going to be about 67F. By the end of April it will be at least 10 degrees warmer.