The short version

Jumpstash does not collect, store, transmit, or share any personal data. There is no Jumpstash account, no Jumpstash server, no analytics SDK, and no telemetry. Your two-factor codes live in your device's iCloud Keychain, end-to-end encrypted by Apple, and are only accessible to your own Apple ID's devices.

If this page disappeared tomorrow, the answer would still be: nothing leaves your device.