If it is a direct 'thru flight' they need to:
1) notify the FA before they deplane so that their thru count is right
2) FIND and notify the gate agent responsible for boarding the incoming passengers. they are the only employees who can easily mark a passenger as 'deplaned'. the agents at the gate counters usually just shrug their shoulders or do nothing and say you are all set.
otherwise there may be delays, recounts, etc.

If it is a connecting flight, you should:
1) notify and hand the connecting boarding pass to the agent working the connecting gate. prevents them from holding the flight looking for and making PA announcements in the airport and on the plane for your CP's name

Also the CP of course can not check luggage to the hidden city.