Dooer, very interesting question.
First, let me assure you than I am not speaking about an experience with a "great" player and my comments will be limited to hitting.
I tend to think that private lessons with a highly qualified pitching coach could be very valuable for a lot of reasons.
When our son was 14-15, we tried private lessons with two different hitting instructors. One was a former major league player and one was a AAA player who battled long and hard to try and get that last step. Both were extremely qualified. However, each had very different and individual styles. After a number of lessons with each...and more than a fair amount of $$$$,
, we decided to forego the private lessons. Our son could not successfully adapt and make the changes that they were seeking. We made sure he kept at it more than long enough. He wanted to stop the first day.
So, he ended up essentially where he started...seeing the ball long
, being a contact hitter with gap power, rarely striking out, all combined with some very good hand eye coordination. The reps came from my throwing hours on end of BP with the shoulder problems to show for it.
IMO, it is critical for hitters to get reps and a lot of them. If they have solid coaching and a good base from which to work, then I question the need for private lessons unless they are of the type where the player can adapt and be successful with changes introduced. There are many variables in hitting instruction and approaches.
Interestingly, when our son got to college, the hitting instructor would have him make adjustments during nearly every swing/at bat in practice, and he could do it, immediately. But he always ended up back with with his strength.
I guess I would come down this way: if your son is struggling with the basics or has flaws in his swing, then private lessons would be important. Once his swing is established, reps are more important than having private instruction. At least that was our experience.