Karen, yes ... I can't imagine having a pool without it being screened. I've heard and read of instances here in Fla. where people (whose pools aren't screened) finding gators, snakes and even black bears in their pools!
I did find a bat in our screened pool area a couple of years ago; it likely got in when my husband had the door open to the yard for the dogs one evening. Luckily the bat didn't drown and hubby was able to scoop it out with the long handled net.