When choosing compressor you need to consider if it will be able to constantly provide desired pressure and level of air. Small airbrush compressors small compressors boast amazing pressure Air Compressor For Laser Cutting figures but actually poor flow rates adequate only for airbrush pistols. Also if small compressor can push enough air it will likely overheat in short while.
Our laser cutter was included with little membrane pump (Smallest upon the picture), I really do not know very well what specifications it has but blast of air was very weak. It is enough if all you have to to do 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 container which can be filled to 58 PSI. It were able to give us constant 1 bar (14 PSI) of pressure, but it had to operate constantly 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 concern with overheating since compressor engine is not on at all times. Using reserve air in tank we can also drive up to 3.5bar (50psi) for brief jobs.
The best setup in my opinion would be compressor rated to >6cfm, large 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.