Transcription runs on your machine, using a speech model Studio downloads once, about 1.6 GB. On an Apple silicon Mac it listens to a minute of audio in a few seconds. Cutting, previewing and rendering are local too, using a video engine bundled with the app.
What does go to TubeSpanner is the transcript and your script, so the edit can be planned against what you meant to say, and the shot list of the finished edit, so it is saved with your project. The video file stays where it is.