That's correct - all you have stored on your own computer is the tiny cookies which remember your login details. The RHP server stores everything else and just sends down temporary pages which disappear when you delete the browsing history on your own computer. I'm sure you already do this kind of housekeeping to keep your hard drive free of such leftovers.
If you're looking to save copies of completed games on your own computer, that's a completely different question. Since I've never felt the need I can't assist with that.