I have made some progress, at least I am experimenting with some ideas.
But as a beginner still I would appreciate some help..
The userscript (widget) will have to read cookies to determine if the user is logged in.
For example, my website is example.com
When a user is logged on, example.com creates a cookie named 'PHPSESSID'
which expires when user is logged out.
While the cookie is active our userscript (widget) which is a greesemonkey script will remain active.
Now, guess a user is browsing codingforums.com with our widget showing at the bottom.. whats the code to read the cookie 'PHPSESSID' that was set by example.com?