Do I need to stay connected to the internet to keep my games connected from other platforms in Galaxy 2.0? Can't really test it out myself since I don't have the beta yet. So I figured I could ask you guys.

Not sure if I am totally getting what you are asking, but I will give it a shot:

That depends ;o)
Your local Galaxy libarry is definately not internet connection depend, but as Galaxy integrations usually start the games through clients, it wholly depends on whether the client and/or the game supports it. I guess steam games will work as there is an offline mode for steam, and there is one report that (some?) origin games will start.
But in the end you have to try it out yourself.

Thanks! That answered my question. Still gotta wait some time until I can try it myself. Signed up on the 29th.