A private instructor needs to have a good eye. But if he wants to have a great eye, and really see what's going on mechanically, video analysis is a necessary tool.
A batter's swing and a pitcher's delivery are two of the fastest actions in sports. When the front foot hits, the rest can happen in the blink of an eye. Without the use of video, even the best instructor can miss the most important mechanical flaws.
I use high definition video analysis during each of my lessons - on the spot with the player during the session. Each player's videos are then emailed to the parents and posted online so the player and his family can review his mechanics at home along with mark ups and commentary on what adjustments need to be made.
Here is my Coaches Eye Video Library.
And for other baseball videos and older analysis, my YouTube Page.
And here are some examples of pro ballplayers doing things the right way.