When choosing compressor you need to consider if it will be able to constantly provide desired pressure and volume of air. Small airbrush compressors small compressors boast impressive pressure figures but really poor flow rates sufficient only for airbrush pistols. Even if small compressor would be able to push enough air it will likely overheat in couple of minutes.
Our laser cutter came with small membrane pump (Smallest on the picture), I do not know what specifications it has but stream of air was very weak. It is sufficient if all you want to do is engrave, although even then we got substantial 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 run all the time and overheat badly within 5 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 can run at constant ~2bar (31psi) without fear of 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 short 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.