Whatever the scenario it becomes necessary to make an emergency descent. Do you:
1. Mayday ATC and inform them you need an E.D. and wait for a clearance?
2. Hurtle on down and THEN inform ATC what you are doing?
Concerning the emergency descent only:
1. Start emergency descent per AFM
2. Change transponder to 7700.*
3. Notify ATC, declare Mayday.
4. Checklist.
The above after donning the O2 masks obviously and some operators may have numbers 3 and 4 switched.
I've had four rapid decompressions, two in the 727 and two in a Lockheed Jetstar. Those were the procedures I followed, no issues with ATC about starting the emergency descent without notifying ATC before starting the emergency descent.
Follow the old adage of 'Fly first, talk later'.
Once I heard an American Airline aircraft that had to make an emergency descent, the first thing we and ATC heard was the FE reading the emergency descent checklist, pushed the wrong mic button. Only after the FE finished the checklist did they call ATC.
* In a radar controlled environment, 7700 will get the air traffic controllers instant attention.