So even names have a gender? So I assume male names have male gender, and female names female gender. But what about unisex names such as טל?
No, Shem is a masculine noun. It doesn't matter what the name is.
A girl can say: שרה הוא השם הפרטי שלי.
It literally means personal, but is referring to your first name, which is personal
Actually it would be first name unless we are talking the English of Shakespeare and Chaucer.
I wrote private name as in an undercover name or pseudonym. I wonder how such a concept would be expressed in Hebrew.