The two TCAS's will coordinate a plan of action with each other that keeps the planes apart... but that theory depends on BOTH AIRCRAFT ADHERING TO THE RA. Obvious problem: guys with an emergency descent may or may not do everything that the RA is telling them to do. What we're told as pilots is that no, the software does not have time to "realize" that one plane isn't doing what it's supposed to & then formulate an alternate plan. To avoid false alarms (like the guy zooming up to 10, while you're level at 11), the software waits until it's pretty sure that there really is a conflict, which means that you no longer have time to "see" if the other aircraft is doing its thing.
There was a crash over Europe (I think) a while back, where TCAS told the jets to do one thing (X goes up, Y goes down) but ATC had the opposite plan in mind & was giving it over the radio about the same time. Sure enough, one jet did what ATC said, the other did what TCAS said, both did the same thing, and they collided. Outcome, we're now told that TCAS takes precedence over the controllers, since you're guaranteed that both TCAS's are one the same sheet of music (they coordinate with each other over Mode S), but both aircraft may or may not be hearing the same controller, etc.
The solution to your scenario is straightforward enough, but probably not in the checklist: the emergency descent aircraft should switch his TCAS from TA/RA to TA only. This means that they don't get an RA and during the "coordination" the other jet's TCAS does what it needs to do in order to avoid the collision WITHOUT any action by the emergency aircraft. Of course, a rapid decompression isn't the ideal time for guys to be considering stuff beyond what's in their checklist, and I don't know if it's common to include "Transponder - TA only" as a step in the rapid descent checklist. It IS in our engine shutdown in flight checklist (and I understand that this is pretty common) -- if we're down to single engine performance, you don't want somebody's TCAS calculating a mutual solution that we don't have the power to give perform!
If you're screaming down with the cabin altitude horn blowing & your ears popping, wearing an O2 mask with the adrenaline flowing, and then you get a the "monitor vertical speed" RA, that's another oh %$#& moment in a day that's already going pretty crummy.