I do not understand Googles implementation of oAuth, but it works. However there is not necessarily an exchange of tokens on each connection. As I understand it Google are issuing token for a long period of time (Months) and these are simple sent by Thunderbird when requested in much the same way a password was offered when required.

However your mention of privacy makes me wonder if you have disabled cookies. They are required n an oAuth process.

Sorry, my question was not clear. I know that OAUTH works. But there is 3rd party app for google created by Mozilla that is connected to every gmail that use OAUTH, i know this is nessecary for this to work with gmail. My questions is, does creator of this app (Mozilla) have access to every google gmail that use OAUTH or its not the case because the tokens are stored locally?

Mozilla will know nothing about the interactions you have with google. It all occurs between your local application and the google sever. Just as Thunderbird knows your password for a non oAth connection, but Mozilla has no access to that either.