Node is flagged, node author comes back to the site, how to show how many new flags since last visit?
To answer your posted question (title), you'll need to extend functionality. 

By default, the core comment module &amp; flag module have no default mechanism to store comment/flag view history. You would need to do something like:


Build something like the Drupal core history module that tracks a user's comment/flag views 
Build a simple aggregate comment/flag query based on a user's last login/access time
...or maybe find a module that offers this functionality (not sure it exists and module recommendations are off topic here)