Is it possible to have full read/write permission but no communication with the outside world?

Presumably, this would involve denying the ability to send emails and denying various forms of network access.

The API that launched today is just a backend API, so by definition your email data is accessible to the application on their servers.

You may be confusing how Streak works, where we have a browser extension that runs inside your browser, but our backend is the one doing the snoozing.