I tihnk I'd be looking in article 422 rather than 430. This seems more like an appliance than a motor, but it is a motor operated appliance so there are a bunch of cross references in the code.
Do the instructions demand an individual branch circuit?
Is the nameplate current rating less than 10 amps?
If there is no nameplate, then its off to article 430.
A 20A circuit should easily run a 1 HP motor. Two half HP motors should be about as easy.
I think 422.10(A) if you consider both GDO's to be one function, or 422.10(B) if these are separate. Section 422.10(A) takes you to 210.23. So you couldn't have lights or non-fastened-in-place equipment on this circuit.