मुझे अपनी किताब दो means "Give me your book" because the subject of the sentence, albeit implicit, is "you". So, saying "मुझे अपनी किताब दो" is essentially the same as saying "तुम मुझे अपनी किताब दो".
A sentence like "I want my book" (i.e. where the personal pronoun (here "my") refers to the subject (here "I")), however, can be translated as either "मुझे मेरी किताब चाहिए" or "मुझे अपनी किताब चाहिए".
I wrote मेरी किताब दो and it was marked wrong. Been speaking Hindi for as long as I can remember and never needed to say मुझे before. Is it really necessary?
The intended receiver is almost always implied in spoken language. As an isolated, written sentence though, मेरी किताब दो doesn't really indicate who the intended receiver is.