First off, lets begin by acknowledging that this is NORMAL. Just because a performer wants to be looked at or listened to while they are showing their skills or character, this does not mean that he/she wants to connect on a personal or social level with others.
Secondly, I encourage my clients to start off by ACCEPTING that this dichotomy may occur, and to avoid fighting it. For example, a singer may feel like he/she is portraying a role when performing; that the equipment, the stage, the performance outfits and the lights are all some type of "mask" behind which he/she hides. Coming to terms with the fact that this "mask" comes off only to reveal a different but equally valuable persona, will relieve the stress of feeling like the confident performer has to exist in day-to-day life as well. A person learns to accept both sides as equally genuine and powerful without feeling like there has to be an amalgam of the two, or that one is better than the other.
Thirdly, we work on identifying and EXPLORING the extent to which shyness affects the performer. If it's shyness which translates into anxiety while on stage, then we work on stage fright management skills. Shyness is one thing; debilitating fear of being negatively evaluated is another!
Finally, my clients and I spend some time incorporating their "true nature" into their artistic IDENTITY. If what one truly wants to do is to perform contemporary dance pieces in an intimate setting, there is no need to push an identity of someone who likes to dance jazz pieces in front of thousands of guests. Of course, one would have to take practical concerns into consideration (how easy it is to turn down jobs nowadays?); however, being able to use one's perceived shyness as a tool that gives them artistic uniqueness rather than an enemy that needs to be fought off, can also pay off emotionally and artistically.