24v from a double circuit outlet
Anytime a dead wire runs parallel to a hot one, it will pick up phantom voltage.   You can see it with a rather sensitive voltmeter like a modern DVM, but it won't do any useful work.   In other words this is an issue with your voltmeter.  Stick a dollar-store nightlight in each outlet and that will eliminate the phantoms.  

The bigger problem is this shared neutral.   Neutrals do not have circuit breakers.  Two hots on the same neutral can overload the neutral, and I notice some scorch marks and heat damage on that neutral wire. 

The only way "sharing a neutral" can work is a special arrangement called a multi-wire branch circuit (MWBC).  Each hot is on a different "leg" of 240V, and the neutral only carries the difference in currents.   Becuase of numerous mishaps relating to shared neutrals, Code now requires you install MWBCs in a way that guarantees they are wired to opposite poles.  Such as using either a 2-pole 240V breaker or two adjacent breakers with listed, approved handle ties.  I can tell you are not to modern Code on that, because you can turn each side off individually. 

The most common mistake with MWBCs is using duplex/tandem breakers, or half-width breaker arrangements which put both breakers on the same pole. This guarantees a neutral overload situation.