"There is your book and mine."
Translation:Il y a ton livre et le mien.
So let me get this straight: le sien and la sienne are his and hers, so they would agree with the gender of the person, but the choice of le mien vs la mienne and le tien vs la tienne is based on the gender of the OBJECT, not the person? Which do you use when it's "c'est" and you don't necessarily know the object's gender?
French possessive pronouns always agree with the object, not the person.
It is absolutely necessary to know the object's gender. If you don't know what a particular mystery object is, you can say:
Ce truc, c'est le sien. / This thing, it's (his or hers). un truc
Cette chose, c'est la tienne. / This thing is yours (whether you are a man or a woman). une chose
But if you are discussing tomates, fleurs, or machines you must use feminine possessive pronouns. Conversely if you are referencing ordinateurs, miroirs, or stylos, you must use masculine pronouns. It is important to learn the corresponding article with the noun in French. Do it early on and you will save yourself a world of trouble later!