Note that everything here assumes you are using individual wires, not cables. 12/2 and 10/3 usually refer to cables but also work as shorthand for "2 12 AWG wires plus ground" and "3 10 AWG wires plus ground". You can't run regular NM cable outdoors in conduit because it is a wet location. And lots of other complications as well.
Aside from the issue of identifying feeds from multiple panels, there are 2 issues with "lots of wires in one conduit":
This is easy. Checking with a conduit fill calculator it looks like 1" will be fine, though there are variances depending on which conduit type, how many ground wires (can be shared between circuits), etc.
This is the complicated part. I believe the neutral in the dryer circuits gets effectively skipped, which means 2 x 4 = 8 conductors. According to 310.15(C)(1):
- 7-9 wires = 70% of 310.16 90 C column
That means 12 AWG = 30A x 0.7 = 21A (more than the 20A circuit you can actually install) and 10 AWG = 40A x 0.7 = 28A (less than the 30A circuit you can actually install, but I am pretty sure (Harper or someone please correct me if I am wrong) that as long as the dryer is actually rated for 28A or less (typical is ~ 24A) you are OK.
However, that means you can't add any additional circuits in the same conduit. One more wire gets you to 10 wires = 50% derate! That would mean bumping up wire sizes, which will increase cost considerably. That's really important if you need any additional circuits in each laundry area for any reason.
In conduit, white and gray are always neutral. Green and bare are always ground. Everything else is always hot, switched hot or traveler. There is no functional difference between the two hots in a 240V circuit. Size helps distinguish wires between circuits. Ground can be shared at the larger size. So you need hot 12 (e.g., black), 12 white, 10 hot x 2 (same color ok), 10 white and a 10 ground.
If you run all 4 circuits in one conduit then mark the end of all the wires from each set with a different color - e.g., red on one, blue on the other. That way you can keep the hots and neutrals paired together correctly, which is critical. If they are not paired correctly then they will not work at all on a GFCI, but if you are in a place that still allows new 240V circuits without GFCI then you could have mismatched neutrals between panels and you wouldn't know it - and that would be a big problem.