COSPAS SARSAT rides along on new GNSS (GPS-like) birds from several nations because it's a good place to put it. They move in the sky (which means a beacon without its own ability to determine location can be triangulated using the doppler effect, not possible from a Geostationary communications satellite), there are lots of them, and they cover the whole Earth (the weather sats used for COSPAS SARSAT today are focused on areas where people care most about the weather, rather than aiming for global coverage...)

The COSPAS SARSAT beacons run at a different frequency to GNSS, they emit bursts of digital data at 406 MHz when active, containing a unique ID for the beacon, optionally (if fitted) its own estimate of its location, and some other data. A global database maps unique IDs to owner/ operators and the nearest rescue agency will try to figure out if this is a genuine alert (e.g. phoning your emergency contact "Hi, is Gupie out on their yacht today? I see, and how many are with them?") and then deploy appropriate resources.

Beacons are entirely passive until you activate them, and then you really want absolutely anybody to know where you are because you want rescuing. "Privacy" feels like a distant problem if you are currently in a leaking raft a hundred miles out to sea or just broke your leg half way up a mountain.