Personal Rating is the rating you see on the PvP tab, MMR is your “hidden rating” you can see it in the scoreboard inside an arena at the bottom-right.
You are matched depending on your MMR and the personal rating you gain or lose is based on the differential between your personal and MMR.
MMR fluctuates a lot faster than Personal rating and makes it easier to match you with people of your skill earlier to avoid gladiators dominating low rated queues for too long while simultaneously slowing down the personal rating climb to ensure it takes a minimal amount of time to actively reach the personal rating you should be at.
It’s not a perfect system but it’s a system, literally.
It would be a good solution to deal with people abusing the system, but it wouldn’t fix the issue of everyone else being punished if a leaver does happen.
Some might even abuse it to just tank their CR and MMR to play at lower rating to dominate and get conquest/great vault cap faster, ruining it for lower rated solo shuffle.
It’s not an easy problem to fix, but it should definitely have been figured out a while ago when it was reported in beta.