GitHub stays the system of record. Gitsu has no issue database of its own: it reads and writes your boards through the GitHub API with your own token.
To open boards quickly, Gitsu caches recent GitHub responses on your device and reuses them for up to 24 hours. Sign out clears the GitHub token and cached data from the device. In the Mac app, your GitHub token sits in the macOS keychain.
A few paths do pass data through Gitsu's servers. The Web App sends GitHub requests through a Gitsu proxy that attaches your session's token. Paid AI suggestions send the issue's title, part of its body and example issue titles to Gitsu's API and a hosted AI provider. The data-handling page and privacy policy cover each one.
Only if you leave analytics on. When you first import your boards, a Help improve Gitsu checkbox sits under the list, and whatever you leave there is saved. In the EEA it starts unticked; elsewhere it can start ticked, so untick it if you don't want to share.
Analytics are anonymous, aggregate usage events and never include your issue content. You can change your choice any time with Share product analytics under Settings → Privacy. If Gitsu can't load your current setting, analytics stay off until it can confirm it.
Nothing. Your issues and boards live in GitHub, not in Gitsu. When a subscription ends you lose the desktop app and paid features, but every issue, comment and field value is still on github.com, and the free Web App still opens them.