How do you make sure everyone in the OR is "in-network"?

I have been reading posts about people who have had to pay fees for an anesthesiologist that was out-of-network. How do I know that everyone in the OR is going to be in my insurance network? 

You can find out what anesthesiologist group they use and call the group. Let them know what insurance you use and just make sure they are sending someone who takes it. I have done that before. Usually more a problem in emergencies but it can happen.

I was under the understanding that if your doctor is in network it is his responsibilitity to make sure the people in the OR are in network. 

I have read the opposite. I have read that it's the patient's responsibility to deal with his or her insurance company and find out if the anasthesiologist in in-network. It seems really twisted, however, because how will I know who comes in and out of the OR when I don't even know what's going on?

It is only the physicians who are billing, they aren't changing whose in there in the middle of your surgery. Doesn't matter if anybody else changes, they aren't billing. You already know your surgeon, and you meet the anesthesiologist before your procedure.