We write "his dog" and "her dog", not "his' dog" and "her' dog".
The words "his" and "her" are possessive pronouns. They don't need an extra apostrophe to indicate possession because the words themselves indicate possession.
A common point of confusion is between "its" and "it's". The word "its" is like "his" or "her" - it is a possessive pronoun, so it needs no apostrophe. The word "it's", on the other hand, is a contraction of two words "it is".
The following make no sense:
1. "The dog brought the newspaper to it's owner."
2. "Its a nice day today."
Sentence 1 reads "The god brought the newpapre to it is owner."
Sentence 2 reads "Something unnamed owns something else (also unnamed) that does something unspecified to a nice day today."