Honestly, to be successful I recommend both. You always want to have a good home training regimen to constantly work on and improve behaviors and keep your dog happy with a job. But school classes are also helpful because they add distractions (other people and dogs, new environment, new tools) that can be hard to produce yourself in a controlled environment. I know I can't produce three to six other dogs that will stay within a workable distance from me at will. And it is even less likely that I can produce dogs that are also being worked with and therefore easier to proof around during starting training versus trying to proof around a random dog in a pet store that might have an owner that wants your dogs to "play" while you're just trying to proof training.
Plus having a trainer to work with gives you a second pair of eyes watching everything you do, and more importantly catching your mistakes.
I was working Kaiju on an agility course and he just kept going past an obstacle he was supposed to be jumping and I just could not figure out why on earth he was doing it. After having my trainer watch me a couple times, she pointed out that I was turning my feet at a different angle than my shoulders and from the tunnel view where Kaiju came out, I looked like I was oriented in a completely different direction - something that I probably only could have seen with a mirror.
So even for those very experienced in training, attending classes usually has a positive impact on your training. I'll be going through all these same classes again with the next dog I get, and I'm sure they'll still benefit me and I'll continue to learn a thing or two.