It depends on the nanny's role in your child's life and, I guess, how busy you are. We did childproofing on an as-needed basis. We believe in teaching the baby not to touch certain things, so we didn't want to cover the whole house, but he took his first steps at six months and wasn't too steady and wasn't hearing every "no". We have hard floors throughout, so I got him a helmet and covered the sharp corners. His play area was in front of the fireplace, so we got bumpers for that. Then we worked with the nanny to determine what else should be "proofed" and when. When he started noticing the outlets, we plugged them. When he started opening cabinets, we locked only the one under the sink. Unless it's something really dangerous, we just wait until he notices it and see how he responds being told to leave it alone. This is his home, so he has the right to explore. Sometimes he just wants to know "What's in there?" We let him check it out and then teach him to leave it alone. That usually works.
To answer your question, I think that it's a joint effort. The parents should take the initiative, but the nanny keeps an eye on the baby and lets the parents know what holds the baby's interest. Because I want the nanny to be more of a personalized babysitter-tutor-extension of me than a surrogate mother, I take her suggestions and make the final decisions. She let us know when she thought it was time to go out and get the 'proofing stuff. As I said, we use it as it is needed. We came home one day, and she had the cabinet under the sink locked.