Can a debt collector for the company I work for call my personal cell phone to demand payment?

It is unclear whether or not the debt is for a personal or business debt; however, I will assume the debt collector is calling about a business debt because you indicated they called you on your personal phone number and that your personal business is not their concern. Unless you personally guaranteed a business debt, it is not your personal obligation.  If in fact the collections agent is calling concerning a personal debt that you owe, you should send them a cease and desist letter indicating that you are not to be contacted about the debt. They can continue collection efforts, but they may not call you or contact you by phone after they receive such a letter. It may help if you clarify exactly what's going on to answer your question further.