Which possessive pronouns should I use in "I'm not a person who..."?

Depending on the context, his or her or their could refer to someone other than I.

I'm not a person who screams out his achievements from the rooftops. I just believe President Trump is due some credit.

I'm not a person who screams out her achievements from the rooftops. I'm just a quietly supportive parent to my daughter.

I'm not a person who screams out their achievements from the rooftops. Those guys get enough publicity as it is in any case.

The use of my makes clear that the speaker is referring to their own achievements thereby removing any ambiguity that could arise.