Yes.
I have multiple consoles , and 1 is set to private. It does not appear on the map. My sensors that are private are for my server room at the office. There is a little bit of a strange thing when I click on the map with that console selected on Ecowitt.net (Understand you have to be logged in as the station owner) and it goes to my area where that is, but no pin or marker. So for anyone else in the general area , just zooming in, or zooming out , won't reveal the location for someone else doing the same thing who may be looking for other sites.
In other words, you can't find it