How do I allow users (registered on my site) to use my firefox addon

I created a Firefox addon using the Greasemonkey script compiler at arantius.com/misc/greasemonkey/script-compiler. The addon receives data from my server and displays it on third-party sites. Now I want to restrict access only to authenticated users. What is the best way to do this?

Users have an account on my website (made with Drupal 7) that sets a session cookie when they log in. So I tried to get my addon to read the session cookie, send it to the backend, which then checks the login status.

But since the javascript of my addon runs on a third-party site, it does not allow access to my cookie. I get this error:

Error: < http://de.wikipedia.org > wurde die Erlaubnis für das Lesen der Eigenschaft XPCComponents.classes verweigert.

(en: "Error: Wikipedia was not allowed to read XPCComponents.classes property")

Despite security concerns, I tried using unsafeWindow, but the code still doesn't work:

try {
    unsafeWindow.netscape.security.PrivilegeManager
    .enablePrivilege('UniversalXPConnect');
    var cookieMgr = Components.classes["@mozilla.org/cookiemanager;1"]
    .getService(Components.interfaces.nsICookieManager);
}
catch (errorInfo)
{
    alert(errorInfo);
}

Now I know that cross-domain cookies should not be possible. But how can I get around this? How other add-ons perform authentication (for example, Delicious Bookmarks, StumbleUpon and many others).

Any tips or pointers would be greatly appreciated.