What you have is a multi-wire branch circuit (MWBC). Properly installed, they are perfectly acceptable and legal. The key point is that the two hots must be on opposite legs in the panel. Because of the way A/C works, two legs are out of phase by 180 degrees so the neutral carries the difference in current between the two legs. It's biggest advantage is cost savings because you save one [neutral] wire for every two circuits.

Frankly, I'm not overly fond of them in residential setting because people don't understand what's going on and there's the potential to get zapped. It's critical to shut off both breakers when working on a MWBC. The rules say that the breakers should be tied so that both are turned off at the same time but that doesn't always happen.

Let's say that you have a room where a wall receptacle and overhead light are opposite legs of that MWBC and you decide to replace the receptacle. Unaware it's a MWBC and because the idiot who installed it didn't tie the breakers, you turn off the breaker for the receptacle and test it. It's dead and you start pulling things apart. Because you want to see better you flip on that overhead light. The receptacle will still test dead but as you're handling the neutral you hand brushes up against the grounded metal box. You get zapped. It's not likely to kill you but it's going to hurt. Change the scenario slightly where there's a table lamp plugged into the wall receptacle and you're up on a ladder working on the ceiling fixture. Get zapped slightly, naturally jerk back, loose your balance, and fall off the ladder.

So make sure those breakers are tied.

"The receptacle will still test dead but as you're handling the neutral you hand brushes up against the grounded metal box. You get zapped."

While there are risks to MWBC circuits, this on e is sort of a fantasy.
You are not likely to feel anything nut the most minor of shocks, the path through you and ground is not going to be a lower impedance than an intact neutral.

The bigger requirement is that neutral in MWBS circuits be pigtailed so that disconnecting a devices does not interrupt the neutral to other loads.

Thanks for the advice and definition of the terminology. So I have a question about CGFIs in this situation (which is what precipitated my first question in the first place), but since it is a different enough topic I have posted it separately.