Do I need to separate my recyclables or do waste management companies do it anyways?

I've become a stickler for recycling, but my boss said something the other day that made me question as to whether or not it's worth the effort. He said we didn't need to separate the recyclable items at our restaurant because the waste management company does it automatically. He told me that within the past 10 years, companies have begun separating out recyclable items upon receiving trash. He explained that he had seen a whole documentary about it and that they've become really efficient with separating out plastics, metals, and other recyclable items and sending them to the appropriate locations to be recycled. This is making me second guess as to whether or not it's even worth my time and effort to peel off labels, rinse and recycle. Is this just true for the company that manages the trash at the restaurant or are all companies doing this?