Can a Nurse Practitioner Who Has Been Trained to Inject Botox and Juvederm Do So Without a Collaborating Physician?

The answer varies state by state, but to give you some reference, in Wisconsin NP's still need supervision.  That level, unfortunately, is way too loose.  The supervising physician can be of almost any discipline (core disciplines vs family practice, er, anesthesiology, psychiatry), and practice only within around 100 miles of the injector.  That's obviously an accident waiting to happen.  One other consideration is that many of the vendors will not dispense to someone who is not a physician.  So you won't be able to even order your botox or fillers.

In Virginia, nurses who have been trained are allowed to inject Botox and fillers.  I think you should ask yourself who you would want to inject you?   Plastic Surgeons undergo years of training in order to be able to practice medicine and surgery.  This includes at least five years after medical school.  Nurses are able to fully practice after one year of nursing school; the nurses injecting usually did go to a weekend course, but is that really what you want?

I am not a lawyer and every state has its own laws governing this. What I can tell you the definition of "training" does not confer expertise on any injector, nurse practitioner or physician. There are folks who are "trained" that spend an hour with a nonphysician sales rep who shows them how to inject Botox or Juvederm in a crash course. The company obviously wants to put their products in the hands of as many injectors as possible. Then you have physicians like dermatologists and plastic surgeons who "train" for several years and learn the anatomy, the nerves, the arteries, the veins, the muscles of the face in detail and are supervised by board certified physicians that teach them how to inject and they perfect their skills over years. 

The two are not equivalent.