Use Private Browsing Window (or add option to do so)

If I am logged into FB in Safari and a session is saved, but someone else wants to login to the Messenger app on my mac, they are unable to verify their account because my FB session is still active and they'll see a "You must logout to see this page"-style message in FB: http://snpy.link/jPUcLv

This happens when their account has two-factor auth turned on.

Using private browsing could avoid this.

(Though it occurs to me that this would make things like 'remember me' not work. Perhaps there is a better solution to the issue?)