Tabhunter saves the following data in local storage:
- search-pattern
- show-audio checkbox
- whether to hide discarded tabs
- whether to close on the go button
- the preferences
That's it. Tabhunter doesn't save anything else, and doesn't communicate any data to other machines. With the only exception that if you have somehow mapped the volume containing your local-storage directory to another machine, that data will be transmitted remotely.