I wish that it was more prevalent for people to just assume and know that any website, regardless of whether or not it has a chat function, has the ability to record what you're saying (read: typing). Everyone out there should act like everything they type into their web browser has the potential to be seen by, at the very least, the website they're typing it on and, at worst, by everyone online. If you really need t…
If the site is using SSL, there is a reasonable expectation that only the server will see that data, not the entire world.
Edit: In the worst-case, as per the GP's point.