As parents we want our children to be healthy and happy. It sounds as though this young lady is both. One of the things we always want to remember is that our children develop in many different ways, physically, cognitively, language and emotional/social. For three year-olds their social development is typically that they like to play among, not with, other children and they do not share their belongings willingly.
To increase her comfort around other children you may want to start play-dates where you and a friend get together for coffee for a short time and allow your children to be around one another so they get used to playing "around" another child. First have them come to your home for some time and then as they get used to one another, you can try other locations such as a park or an ice cream shop for a treat, the zoo or the other child's home. Since it appears she is very sensitive, ease into this slowly. Allow her to form a bond with one other child before expecting her to play or make friends with many others.
Relax and help her have fun one on one in her home environment. We are not all the same. Many of us have learned how to get along in this social world we live in but honestly would still rather be alone when given the opportunity.
