Here's the long answer: Social anxiety and performance anxiety (stage fright) are in the same family of emotional and somatic experiences involving the preoccupation with how one is being perceived by others, heightened fear of being humiliated and a series of physical symptoms such as dizziness, shakiness, rapid heartbeat, sweatiness and overall discomfort. In most cases, performance anxiety does not require the existence of social anxiety. It is very common for people to struggle with tension and anxiety when they have to make a public appearance but they can be very comfortable in other types of social settings, completely undisturbed by such feelings.
It is also possible to have social anxiety disorder without having it interfere with the experience and quality of a performance. Many performers, those who often transform into bursting energies of confidence, experience this type of anxiety before and after the performance, when they are often expected to interact with audience members and people from the entertainment industry. Throughout the duration of a performance, a performer may feel more secure, present and comfortable being the center of others' attention. As soon as the performance is over, the same person may begin to feel a sense of pressure stemming from increased expectations, negative thoughts about how they are perceived and discomfort associated with social anxiety. It is a common paradox experienced by many performers.
The key to managing social anxiety during a performance is to bring attention to the performer identity. Performers take on a different persona when on stage. They embody the stage character and this parallel identity often becomes an additional source of confidence and a safe detachment from the non-performing self. When someone already has social anxiety, they are used to feeling vulnerable in their most natural state; when they are themselves. However, the state of performing allows for the person to become someone else, or perhaps to access another version of a deeper self, and to interact with others through this temporary persona. Accentuating this antithesis can become a source of emotional safety that protects the performer from the distracting effects of the social anxiety he/she experiences as their non-performing self.