The app sends nothing on its own except Claude Code itself and the update check. Everything else waits until you turn it on. The full list, and how it was checked, is on Security and privacy.
The vault
Tokens you save in Connections are encrypted on disk with a key macOS keeps in your Keychain. The app shows their names, never the values.
Updates
The app checks GitHub Releases at launch and every 6 hours and installs only when you say so. Turn the check off in Appearance, Check for updates automatically.