Who owns a conversation? This treads on the edge of two issues in my mind.
Firstly that of shared creations. How do we negotiate the creative input and output of a common endeavor?
Secondly, that of the nature of creativity. A mere conversation can be perfunctory. But over time it can become a dialogue or even banter. Written down it can be considered epistolary. At what point does it become creative, especially over the threshold of exposition and monetary value?
It is creative when it has original content. In terms of rights, if it were recorded or transcribed and then distributed, it should be seen as the joint creation of all its participants. I’d compare it, in music, to original improvisational jams; normally these are co-credited to all musicians heard.
Erving Goffman, who is considered one of the originators of conversation analysis would agree. His writings are good for illustrating how conversational roles and outcomes are co-constructed, even as a conversation plays out. He developed a system of annotation which helps show how people employ ways to do this on the turn of minute fractions of a second.