The band stores readings (context, cost, rate limits, the cache clock, running subagent types) in session state; see "What it can reach" in the README. It makes no network requests. It runs two processes: a tail of this session's own transcript when loaded mid-session, and the costCommand you configure.
If you find a way it could leak data, please report it privately through GitHub's Security → Report a vulnerability on this repository rather than in a public issue.