Does anybody run a pistol on their chest rig or plate carrier?  What are the advantages or disadvantages of doing so?

I don't have mine on my vest. I thought about it but having an AD would end badly.  I just think its a bad idea.

People I've known that wore it on their chest did it because they spent most of their time mounted, drawing from the hip or thigh seated can be a bitch. Obviously its not the safest, being pointed either down at your groin/legs while seated or at whomever happens to be at your side depending on orientation. Its a measured risk I suppose.

I prefer not to have anything getting between me and my mags.  Keep it on the thigh or hip.

Never been a fan of chest mounted weapons or Shoulder holsters, .... says here take me when in close with an adversary.

I have a Tac force pistol holster chest mounted on my 5.11 vest. Seems to be comfortable and makes for fast presentations. I am far from an expert on the subject however.

In my opinion, pistol should be on the belt. If you're in a vehicle seat more often than not, chest mounted may be better.

I have the Blackhawk Serpa holster. It goes on the side of my GI MOLLE assault vest where a holster normally goes.
Holster

Will also add, I have another MOLLE vest with a leg extension and a MOLLE holster, and the older enhanced  load bearing vest (love these), the type that uses the pistol belt, with a leg extension and M9 holster.
Have a custom vest with build in holster on the chest, but it seems rather loose, or should say the vest material isn't stiff.
And then the old GI H type suspender setup, with M9 on the belt. The suspender setup is probably the best basic rig ever made. Between it and the load bearing vest, you can add most anything you need, even use a butt pack as a small backpack and be good for a few days, at a fraction of the cost of newer stuff.

A holster on body armor isn't half bad if you're in and out of vehicles or confined spaces a lot.  The most common configuration I've seen was to have a holster mounted at an angle.