When browsing the GameSparks support site (e.g. this very page), the website still shows the "Sign In" and "Register" buttons in the upper-right corner even if you are already signed in. There is no obvious way to tell if you are signed in.
The sessions across the sites (support, portal2, docs) are maintain in different ways - cookies, WebStorage API, perhaps others. Ultimately, visiting portal2.gamesparks.net is the best check.