The 240vac power coming into the house is coming in from a transformer. Each of those hot leads is coming from opposite ends of the secondary of the transformer coil. 120vac is sort of a phantom, as the power goes through the transformer coil, what they call neutral is tapped off of the middle of the coil so that measured from there to either end, you get 120vac. But, a property of a coil like that is that when one end is high, the other end is low on the sine wave, which makes the middle half-way. In a MWBC, each hot lead is connected to one end of that transformer, so, when one line is at the high end of the sinewave, the other end is at the low end. If you're running power entirely on one branch, the power goes out on one hot and returns on the neutral, so each wire has the same amount of current on it. But, if you now start to pull power off of the other branch, what is returning on the neutral from that side is out of phase from the power on the former branch, and they cancel out, so essentially, the neutral is never carrying the combined power...if it added, the neutral conductor would now be trying to carry up to twice the current the wire gauge in the cable was designed for. Because the neutral is half-way across the coil, it functionally makes it like two transformers, and remember that the phase on each end is opposite of the other, so when one hot lead was positive, the middle (neutral would be negative)...the other side of the coil would be negative, but RELATIVE to it, the neutral would now be positive...the positive and negative values (when the current is the same) totally cancel each other out, but as the power used in each branch normally isn't identical, it could go from 0-100% the same, the neutral won't ever try to carry more than the hot from its associated circuit.
A downside of a MWBC is, if you lose the neutral connection, you might end up seeing 240vac where you were only expecting 120vac.
Because the neutral can be carrying power from either branch, when you trip a breaker, you expect you've cut off all paths for the power...but, if both branches don't shut off at the same time, the shared neutral could easily still be having power running through it in the box where you have the other branch shut off...
What complicates this a bit more, is that they bond neutral to ground, so you may not be able to measure a voltage on it, but that doesn't mean that there's not still the possibility of current running through the connection. The ground is there to offer an alternate path for power when there is a fault, to trip the breaker or blow a fuse...it should never be used as a current carrying conductor in normal operations.
Did that help, or confuse things more?!