> the response is managed and coordinated by the US military.
Who is responsible will vary around the world. If you're at sea it will likely be some sort of coast guard or possibly even a navy, but on land it's often a civilian agency charged with rescuing people who've got into trouble far from civilisation or even handed off to volunteers.
COSPAS/SARSAT (the satellite technology that makes these beacons work) is a rare example of Cold War co-operation on technology for peaceful ends. Hence why SARSAT makes obvious sense in English (Seach And Rescue SATellite) while COSPAS is Russian.
Before all this starts, there are analogue radio beacons. Those are pretty dumb, you can still find them in some personal planes flown by pilots too cheap to upgrade - they just play a siren tone on a fixed reserved frequency when triggered. If somebody is tuned to the reserved frequency and close enough, they can hear your beacon, with some pretty easy technology a group can triangulate and find the emergency. Might save your life, if people are searching and if you can survive maybe a day or two while they look.
COSPAS/SARSAT says wait, we have satellites, put a radio "mirror" on a satellite at that frequency, and now we don't have to search any more. We sit at the ground station for the satellite, we hear distress beacon sirens from existing beacons on the ground, we can use mathematics to work out based on where the satellite is and the Doppler effect where the radio beacon is in a few minutes.
That works up to a point, but there sure is a lot of other junk that's broadcasting (intentionally or not) on the reserved frequency, and you've got no idea when your rescue helicopter scrambles are you going to find: A small plane with no filed fight plan and six badly injured people? One lone hiker who is just lost in the hills? A pile of garbage somebody threw out which happens to include a radio beacon with just barely working batteries? Could be anything. Almost always it was nothing. That's expensive and bad for morale.
So, modern COSPAS/SARSAT beacons are digital, and the satellites are now smarter. The beacon has a unique identifier and if fitted with or connected to a suitable receiver (very common on boats and somewhat common elsewhere) it can transmit its GPS co-ordinates.
This means now the base station isn't trying to guess if this is a beacon with a low battery or a garage door opener, they know instantly it's beacon #4843931 and it says it's fourteen miles off the north coast of Scotland. This cut false alarms by an order of magnitude at least, and reduced search time to almost zero. False alarms are still the majority of all alerts though - emergencies don't happen that often and humans are lazy and incompetent leading to false alarms.
The current work on COSPAS/SARSAT is return link. Right now you trigger the beacon and then... maybe you get rescued. But that's not great emotionally and we know in some emergencies hope is key to survival. Is anybody coming? If you're clinging to the wreck of a yacht ten minutes can seem like an hour, surely the beacon you triggered just isn't working? So return link is a mechanism by which GPS messages are used to tell your beacon "Received. Rescue coming" and perhaps some day more complicated messaging like "Cannot Land. Ground team on its way to you. Stay here".