'Private' generally means you don't want to advertise that you are looking for a new team. Which is why in tryout notices it normally says something like private and confidential tryouts available.
We also have them for those players who couldn't make a tryout date because of conflicts (such as other tryouts, still on vacation, etc). Especially since everyone tries to have their tryouts conflict as much as possible.
I have not seen a player request one because they perform better in a private tryout, It may have happened however the reason given to us was one of the above two points.
I could see for a pitcher or catcher, the advantage of a private workout. On our current team each girl who wanted to pitch, after warming up, threw 5 or 6 pitches. In a private workout there might be more time for the individual I imagine. I like group tryouts because I think that my dd shines in that environment but as she gets more into pitching I might change my mind about that.
Schedule conflicts are a common reason for private workouts. "Hey, I'm interested in my daughter having a shot at your team, but this bigger name organization is having their tryout Saturday at 9am, which is the same time as yours. So, I'd like to bring her to work out for you in advance of tryouts. Hopefully you'll give her an offer that she can have in her pocket when she goes to try out for ___."