Does the anesthesiologist know if you are breathing on your own before he takes you off the ventilator?

Yes: All anesthesia machines have capnographs, which measure carbon dioxide, found in expired breaths. By watching the trace of the capnograph, the anesthesiologist can tell whether you're breathing by yourself or not. You don't have to be breathing fully by yourself to be taken off a ventilator; the anesthesiologist can assist you breathe by manually squeezing a bag until you get full strength back