That is a "Multi-Wire Branch Circuit" or MWBC.
Your setup is grandfathered, but you should update it nonetheless. The two circuit breakers in an MWBC must be a 240V 2-pole breaker. Alternately it's allowed to use 2 breakers with a factory handle-tie, but those 2 breakers must occupy the same panel space as a 240V 2-pole breaker, because it needs access to 2 adjacent spaces so it can get both poles. MWBCs must be on opposite poles (must have 240V between them) otherwise they will overload the neutral.
The purpose of the 2-pole/handle-tie is to assure that when you turn off 1/2 of the circuit for maintenance, you are also turning off the other 1/2.
Otherwise you're asking to get nailed.
But yes, current is moving through the shared part of the circuit! Don't separate the neutrals without doing the above.
If you hate MWBCs / don't want to deal with them, I totally understand. Pull two 12/2 cables to replace the 12/3, and land them on the breakers normally. I have been retiring MWBCs on-sight on my Pushmatic panels, because I can't get handle-ties for the breakers, and *new-stock* Pushmatic breakers have Zinsco disease, because somehow, 康涅狄格州 Electric managed to screw them up.