I'm not a fan of public sweating either, but that's a matter of personal preferences. Germs don't enter your body through your skin, unless you have large
open sores like someone with severe burns. The skin is an amazingly effective barrier to bacteria, viruses, etc. When people get infected is when they put their dirty hands on their faces. This is scientific fact, if you believe in science.
So keep your hands away from your face, wash them when you finish and if it makes you feel safer, carry a small bottle of hand santizer with you and clean your hands between equipment. If you're really overly concerned, take your own towels. Even if you do get germs on your skin, they'll likely be flushed away by the good healthy sweat.
Enjoy your workout and don't worry.