When choosing compressor you need to consider if it will be able to constantly provide desired pressure and volume of air. Little airbrush compressors small compressors boast amazing pressure figures but really poor flow rates sufficient only for airbrush pistols. Actually if small compressor would be able to push enough air it will likely overheat in couple of minutes.
Our laser cutter was included with small membrane pump (Smallest on the picture), I do not know very well what specifications it has but stream of air was extremely weak. It is enough if all you want to accomplish is engrave, although even then we got significant smoke traces around engraved picture.
it was rated for volume of ~2.4cfm and had a 1 gallon tank which can be 
filled to 58 PSI. It managed to give us constant 1 bar (14 PSI) of pressure, but it had to operate at all times and overheat badly within five minutes.
Third and current compressor we got was 6cfm rated belt driven device with 13 gallon container which is filled to ~100psi. With this unit we are able to run at constant ~2bar (31psi) without fear of overheating since compressor engine isn’t on on a regular basis. Using reserve surroundings in tank we can also push up to 3.5bar (50psi) for brief jobs.
The best setup for me would be compressor rated to >6cfm, huge 70+ gallons tank rated to 10bar (140psi) and software air control via Air Compressor For Laser Cutting solenoid valve.