Anyone who does the actual sequencing has the data. I think they mean the final service that does the analysis runs locally in the browser. You still need to trust whoever does the sequencing.
Your identity can be reconstructed from relative's samples. If you submit your sequence its out of your hands forever. The only way to do this privately is to do the sequencing in your own lab.
I sequence my own genome in my own private lab. At least weekly. The rush when the sequencing begins. I almost can't help but sequence all over the place.
