As stated above, it doesn't matter much with private practices. I very occasionally get requests for certain fellowship or residency programs but generally find those practices to consider people who aren't from them as well. Once you have more than a year of experience I wouldn't expect many practices to care where you trained if you're a good surgeon and have a good personality. Medical school wise, I've never had any practices request a certain med school although did have one tell me they wanted someone who was AOA and they weren't willing to budge on that.
Edit: I should note on medical schools, I do occasionally find practices to not consider DO candidates no matter their experience level. Again, this is still uncommon.
As Slide said though, if you want to be in a very saturated area then the prestige might give you a leg up on other applicants. However, if you make it to an initial discussion with the practice then your personality can quickly win or lose the day.
For fellowships, it will be more important to choose based on what you want to do. Cornea fellowships for example vary greatly with some being hardcore transplant heavy fellowships with very little cataract surgery and others that focus more on refractive cataract and refractive surgery.