If the pop just the cat, you'll probably get a downstream O2 code but the vehicle will still run. As others have said, there will be no question the converter is missing. You, your neighbors, and pretty much everyone in town will hear it.
If, however, they snag enough pipe to cut, damage, or remove the upstream converter, your vehicle may throw a fit. Depending on manufacturer, some vehicles will simply revert to base fuel/air maps, set a Check Engine light, and keep going. Others will run like poop, or refuse to run at all. There is the remote possibility of damaging the 02 sensor wires in such a way that they zorch the ECM. Not common, but it can happen.
Brad