First, points of terminological order: a possessive pronoun is any instance of the possessive that substitutes for the thing possessed—that is, in “the car is Sarah’s” and “the car is hers”, both Sarah’s and hers are possessive pronouns. In cases like “Sarah’s car” and “her car”, Sarah and her are possessive determiners.
English has two sets of forms for pronouns used as possessives: as possessive determiners: my, thy, your, his, her, its, our, their and as possessive pronouns: mine, thine, yours, his, hers, its, ours, theirs. (The thou/thee/thy/thine forms are mostly obsolete in modern English, but show a parallel with I/me/my/mine, which otherwise seem like oddball outsiders in the paradigm.)
For non-pronouns used as possessives, we always use an apostrophe and the letter S—Sarah’s, Bill’s, the Queen of England’s, Jesus’—regardless of whether the possessive is used as a determiner or a pronoun:
Sarah’s car is broken
The broken car is Sarah’s
The Queen of England’s hat is purple
The purple hat is the Queen of England’s
However, when a pronoun is used possessively, we never use an apostrophe, regardless of whether the pronoun form has an S in it or not:
Nothing you could say can tear me away from my guy
The girl is mine
And I could be proud that I'm his girl
I would gladly take her place
How could anybody break a heart like hers?
I think the system would be even more strange and irregular if some of the possessive forms of personal pronouns had apostrophes but not others rather than the system which we have, where none of them have apostrophes. Where would the apostrophe go in mine, exactly?