Correct, in that the alert that goes to the satellites would send messages to your agency. If registered in the US, then to NOAA, who will call your registered contacts. If they know your schedule, that may help expedite local efforts. If the boat radios local authorities that you're missing but have the PLB, that too will help. If the members of that team effort work together, it all helps. The Coast Guard boats and helicopters can also search for your homing signal and radio others when you are located.
That's almost, but not quite the process.
The signal goes from the satellite to local user terminals (the network of ground stations) to the NOAA-run US Mission Control Center. Other countries have their own MCC's. There are also MCC's in Russia, France, Australia, Japan and Spain.
USMCC then relays it to the Air Force or the Coast Guard, according to the assigned areas of responsibility in the National Search and Rescue Plan. Generally, the USAF is responsible for inland locations and the Coast Guard for water (there are exceptions: for example, USCG has inland SAR responsibility for Puerto Rico).
If the Coast Guard gets the signal, it will be further relayed to the area, district or sector command centers, depending on who has SAR Mission Coordinator responsibility for the source location. Then the callouts and marine broadcasts begin in an attempt to correlate the signal to an emergency.
It's a great system if you're out of VHF range, but it does take a little while to lead to a launch decision. If you're within range of the CG's VHF network (generally 20 NM of shore), a call on channel 16 is your fastest way to get a response. They'll have your position as soon as you key the mic based on bearings to adjacent high sites, and they can replay and enhance the audio of your distress call to make sense of it.