One of the most evocative tabletop roleplaying campaigns I ever participated in did not use character sheets, no dice were ever rolled, there was no table. There were three players, each portraying a character. The game system was vaguely a cross between Amber and Mage. We didn't know our character's stats we didn't know their spheres or arete, we just described what our characters did, or what they attempted to do and based on previous descriptions about our characters we slowly worked out where they were less than average, more than average, or where they might be superhuman (and how far to push things if we knew the limits didn't apply to us in specific areas).
The stories of this group of characters were narrated one by one, with the GM portraying the majority of the world for a single character at a time. But to keep things interesting for the other players while their characters were "offstage", each player had a roster of bit characters in each other's stories who would often be called on by the GM.
Over the first two sessions, the players main characters didn't interact with each other at all, but by the end of the second session there were a few instances where a bit character from one story appeared in another character's story. For a modern reference consider the suspected "Night Nurse" character portrayed by Rosario Dawson in both Daredevil and Jessica Jones. A few common places also started to occur, and by the end of the third game, the ramifications from events in one person's story would be felt in the story of another.
I think it was about the fourth story before any of the "main" characters actually met each other... but we knew that the stories were interconnected. As we uncovered elements of side characters, we saw that for one player they might be an ally, for another player a nemesis. We didn't maturally assume that the characters we were portraying would automatically be friends, but we might be allies of convenience in some ways, friendly rivals in other ways, or at each other's throats occasionally.
There were no mechanisms in play to determine who reacted in what way to whom. We just followed the story. That's where I'd be interested in heading with this project.