Some login data is sent to the server the user is connected to. For instance, if a post is added as a favorite, the post id, user id, and hashed password/api key are sent to their server over https. This is how they verify what post to favorite for which user. No data is shared between sites.

This login data is loaded from that site's cookies. Danbooru doesn't store login information locally, so it can be added manually by the user and is saved, along with add-on options, within the add-on's storage.