We hear this all the time from parents. Kids who are normally loud, outgoing and showy come to their singing lessons and are quiet, shy, and barely say a word. What is going on?
I have several students that I know both in the studio, and outside of the studio. When we are outside of the lesson they talk nonstop, yell, scream, are total goofballs. There is one kid in particular who goes in a piano lesson with me and then whispers only a handful of words the entire time. Even when we're having fun and being silly!
A one-on-one setting can be intimidating. It's the pressure of performing in a small setting with one person listening for the purpose of correction. But the younger you start the more you get used to performing without the need for stage, lights, and separation from the audience as a comfortable crutch.
The same phenomenon happens with adults students, even active performers. My adult students are always saying, "I did this so well at home!" and "If I could just record my practice and show you you would see that I can do it."
When on the stage, at home alone, or in a group with friends most students tend to open up, let loose and have fun. But if we learn to overcome those feelings of intimidation and to give our best in just a room with a piano, then the sky is the limit for the magic that can happen on the stage!