The neutrals from different circuits should not intermingle. The current that leaves the panel on circuit A should return on wiring associated with circuit A only, not circuit B or C or any other.
Unless, of course, two circuits (drawing from the same leg, I think) are specifically wired with a shared neutral, correct?
Note: My washer/dryer combo is 240V, but I suspect its neutral is shared internally since the appliance likely has 120V circuits/components inside.
When wiring multiwire branch circuits it is important to know and understand what is happing or an overload can and will occur.
Multiwire circuits that are installed to items such as dryers and ranges are not something to worry about as the cable and over current devices are pretty straight forward.
210.4 of the NEC mandate that all ungrounded (hot) conductors must be opened at the same time on multiwire circuits. With a dryer the two pole breaker will do this as well as the breaker for the range. The cable will have the appropriate size conductors and no problems.
When wiring something such as these receptacles the same rule applies. The breaker must open all ungrounded conductors at the same time and the two ungrounded conductors would need to share the same neutral.
What will happen here is one 20 amp 2 wire cable and one 15 amp 2 wire cable will be installed which will result in two neutral conductors of different size. These neutral conductors will land on the same terminal bar in the supplying panel and should they again terminate together would constitute a parallel conductor and then would have to comply with 310.4 and be;
(1) Be the same length
(2) Have the same conductor material
(3) Be the same size in circular mil area
(4) Have the same insulation type
(5) Be terminated in the same manner
Which a 12 and a 14 wouldn’t meet the criteria for a parallel circuit.
Another danger is if both are on the same leg of the panel. This would make the current add on the neutral which will result in an overload and damage the conductor.