The compressor will cycle for 3-5 seconds every time you start the engine. There is supposed to be a 17-35 second delay before it does that. It will run longer if the sensor says the car is low. I hear it every time I start my car. It runs longer in the cold weather.
The compressor should not run with the ignition off. It will expel some air though as Albert said.
The Factory Service Manual has an extensive trouble shooting series of charts. Those charts are very necessary when trying to diagnose problems with the ALC.
98 Riviera SC3800 All stock except gutted air box.
1970 Buick GS455 Stage1, TSP built 470BBB, 602HP/589TQ
Best MPH, 116.06 MPH, Best ET, 11.54