I don't know if it's different state to state, but my understanding is that the perfusionists run UNDER the anesthesiologist's licence.
Which means they're successes and f-ups are on your malpractice.
There's a general attitude of letting the perfusionists do their job, but you need to know how their equipment works and get involved.
It is absolutely a team effort; coordinating pump changes with the surgeons and the perfusionists, to help the patient come "off pump," but it's a very underappreciated importance that you understand the mechanics and physiology of what they're doing and know when to interviene and make changes.
I've always had fantastic perfusionists, and I've rarely stepped in to ask them to do anything differently, but a couple smaller hospitals I've been at, where they don't do much out of the way of straight forward CABGs, I've asked them to run higher flows at certain times when the patient hasn't been ideally cooled to optimize supply/demand, etc.
Point is, the idea of once they're "on pump," that you kick back and disconnect until they're ready to come off is a dangerous oversimplification of the process unless you know they're going to do exactly what you'd want anyway.