When choosing compressor you need to consider if it will be able to constantly provide desired pressure and level of air. Small airbrush compressors little compressors boast amazing pressure figures but really poor flow rates sufficient limited to airbrush pistols. Even if small compressor would be able to push enough air it will likely overheat in short while.
Our laser cutter was included with small membrane pump (Smallest upon the picture), I do not know very well what specifications it has but stream of air was very weak. It is adequate if all you want to do is engrave, although actually then we got considerable smoke traces around engraved picture.
it had been rated for level of ~2.4cfm and had a 1 gallon tank which can be filled to 58 PSI. It managed to give us continuous 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 unit with 13 gallon tank which is filled to ~100psi. With this unit we are able to run at continuous ~2bar (31psi) without concern with overheating since compressor engine is not on all the time. Using reserve air in tank we can also push up to 3.5bar (50psi) for brief jobs.
The best setup in my opinion would be compressor rated to >6cfm, huge 70+ gallons tank rated to 10bar (140psi) and software air control via Air Compressor For Laser Cutting china solenoid valve.