We can see users that are banned with mod permissions at least.
Most of the time this information isn't relayed to the general public (you sorry sacks of...
). As @ Sneeky said, you may see the user's name struck out. Though I do wonder in some cases if that reflects over to users or not. I don't recall every seeing anyone's name struck out until I became a mod last week.
You could always stalk someone's profile you think is banned...easy way to tell is if they're last online time isn't changing and if their recent posts don't seem to jive with what you may or may not remember. Stalking on TPU isn't a recommended pattern, so if you do this, you do so under your own recognizance.
You could do what I did in some very select situations where I was involved in a situation, and a party may have been banned or infracted, I would then contact an admin, usually a higher up admin like Sneeky, Tatty or Erocker. If they feel it is okay to share information in regards to the situation, they'll respond. I cannot condone that as an officially supported method, but if it is something you feel it is important you know then make the effort to reach out to us and make your case.