This wiring method, the so-called "Edison circuit', *is* allowed in at least some parts of the U.S. No problem with it, anywhere in the greater metro Chicago area, for example. I recently re-wired my kitchen, in this
-precise- manner, and the city electrical inspector had no problem with it. He actually wondered 'why so *many* _neutral_ wires?' -- 'cuz I had outlets on opposite sides of the breaker panel on the _same_ breaker. The two hot leads were spliced together in a junction box just before the panel (no splices allowed _in_ the breaker box), but both neutrals ran straight through to the bar in the breaker box. Simpler than running wire to the left-side outlet, and then _back_, and over to the right-side outlet. 'Unconventional', but sound engineering, _and_ code-compliant.
As long as the two hot legs are on opposite phases of the 240V feed, the neutral carries only the _difference_ in the two loads. 'Best case' is that there is -zero- current in the neutral; 'worst case' is that there is current in the neutral that is the same as -one- of the hot legs (where the other hot is supporting -zero- load at that time)
If you use GFCI _outlets_, then there is *no* problem. Speaking from direct experience here.
To use 'downstream' outlets from a GFCI outlet, you *do* have to have a 'unique' hot, _and_ a 'unique' neutral, from the GFCI device to the downstream outlets. You cannot share _that_ wiring across GFCIs.
I'm *not* sure about GFCI _breakers_, having never actually _used_ those devices. I would _suspect_, however, that they work in the same manner -- that you can't 'share' the neutral downstream of a GFCI breaker. "Read the directions" is indicated.