Wait a minute: "John and I's conversation"? WHAT?! Oh gods, where to start.
Unknown to most native speakers, English has declensions. We have three cases: nominative, objective, and dative. No, dear, dative. And, believe it or not, pronouns have to agree with the case! OMG, we are sooo put upon.
Nominative case refers to the named subject of the sentence; for example, I.
Objective case refers to the object of the verb. That would be me.
Dative case refers to the object of a preposition, such as "to" or "for"; that would also be me.
And again, a class of pronouns called possessive: that is, my and mine.
Take a break, dear hearts. Livia knows that this sort of thing is taxing. Now. What is wrong with the phrase, "John and I's conversation"? Speak up, dear. Livia can't hear you. "I's conversation" is a grammatical impossibility, yes?
But what pronoun goes in I's place? Not "me," certainly. RIGHT! "John's and my conversation"! And that's leaving aside the obvious edit: "In the conversation I had with John, we decided..." or the stilted (but instructive): "The conversation between John and me resulted in the decision... ."
Just to keep things interesting, we also have a class of pronouns called reflexive: in the first person, that would be myself. (If it helps, take out the x and substitute ct. The pronoun reflects back to the subject of the sentence.) NEVER, NEVER, EVER say or write
"John and myself had a conversation." (nominative)
"The speakers were John and myself." (objective)
"There was a conversation between John and myself." (dative)
Myself may be used only in a supporting role: I did it myself. It may be the object of a verb or a preposition if -- and ONLY if -- the subject of the sentence is I: I embarrassed myself; I sent it to myself.
Are we clear? I nominative; me objective or dative; my/mine possessive; myself reflexive.
That wasn't so hard, was it? Now mind! Or you'll be thrown from the Tarpeian Rock!