No. The frames are read on the device, the pose is worked out there, and the rep count is decided there. Nothing from the camera is sent up so the count can happen.
What a signed-in account stores
If you sign in, a finished set can save the time, the length, the rep count and a form score, so the log is there on another phone. That is numbers, not the video.
Recordings
Recording is off until you turn it on, and it is part of Pro. The clip is kept in the browser on that phone. Clear the data on the phone, or delete the account, and it goes with the rest.
The free set
One counted set a day is free to start. You do not need an account for that first set.