The passenger should have the option to have this search done in public at the checkpoint. The private room should be an option but not mandatory. Unless there is a strip search, then there is no reason to force this into a private room.
There have been plenty of reports of what the private room search involves and none of it is secret or is SSI. The fact that the TSA seems to be embarrassed to perform this pat down in public says a lot about how inappropriate it is. I don't care if I'm getting my genitals touched with the back of the hand or the front of the hand. Either way, my genitals are being touched.