My perspective is a little different from Bob and Arne, since I haven't had any individual coaching. Instead, I've attended classes given by Fred Asbell, Rick Welch, and Rod Jenkins (twice). Each of these individuals has their own distinctive style of shooting and teaching, and I've enjoyed seeing their different approaches. My own personal shooting style is sort of a combination of what I've learned from each of them, weighted toward Rod's style, as that has seemed to give me the most consistency.
I do believe that any one of them would be able to look at a person shooting and identify the things that person is doing that are causing inaccuracies; things that the person is probably not able to identify for himself. As was mentioned by everyone, that is the primary benefit of an instructor or coach. Most of the things we are doing wrong we have done wrong for so many thousands of shots that it feels right to us, which makes it very hard to self-diagnose.
The main benefit of a coach over going to a class is that I'm sure as soon as I leave the class I start doing some of the old things wrong again, or start doing something new wrong, and a good coach would keep me on the right path better than I can do for myself. Experts at all levels use personal coaches, which means that we never outgrow the need for it.
The main benefit of a class is that you get to meet some new folks, and learn almost as much by watching other people shoot in front of the class and being critiqued as you do when it's your turn. And, as I mentioned, there's more than one way to skin a cat, and you might like part of one instructor's style for one thing and another instructor's style for another. I realize that if you wanted to become an Olympic competitor, a lot of the ways we shoot might not be appropriate, but then some of the things Olympic competitors do wouldn't work out very well in the woods, either.
To be honest, I am the only person out of the group I shoot with that has ever gone to a class or had any formal instruction. I may never be able to shoot as well as a couple of them on a regular basis, although on some days I can. Rick Welch has never had a day of instruction. So you can do it yourself, if you are persistent.