Consider these three sentences:
I have my pen.
I have his pen.
He has his pen.
In Czech they become:
Mám své pero.
Mám jeho pero.
Má své pero.
Did you catch it? In the first sentence “my” translates to “své”. In the third sentence “své” means “his”. But “his” is “jeho” in the second sentence. To complete the cycle, “He has my pen.” becomes Má mé pero.
There is a special pronoun, svúj, that takes over when the possessive pronoun refers to the subject of the sentence, no matter who or what the subject of the sentence is. So in “I have my pen” svúj takes over for the usual múj (and then is converted to the proper form, following the same pattern as múj). Same thing in “He has his pen,” since the “his” refers to the same thing as the “he”.
I’m not sure it improves communication at all, whether it reduces ambiguity in the language. I think it does. When you say “Mary has her pen.” you know right away whether the “her” refers to Mary or to some other woman. I’m liking the svúj.