Local Browser Translator sends page text only to the chat-completion endpoint you configure — by default a loopback (localhost) server. If you configure a remote/cloud endpoint, page text is sent there; that is your explicit configuration choice, not something the extension decides. Translated text and a translation memory are stored locally in the browser profile (IndexedDB and storage.local) and are never uploaded anywhere. API keys you enter are stored in storage.local, which Firefox does not encrypt. There is no telemetry, no analytics, and no remote code — everything the extension runs ships inside the package.