I’ve never done this myself, except as a highly stylized way to write a “after play report”. I was in a campaign where the characters would wander into adventures and share their findings in a Facebook group. (It was kind of a “West Marches” style of setup, so it was pretty important to share this information.) I ended up writing my “after play reports” in-character, as memoirs. It was fun, and I think the others enjoyed reading them.
I have two things to contribute to this thread:
Once upon a long time ago, I was running a long, ongoing D&D campaign, and ended up moving 3,000km away. Playing online wasn’t possible back then, and we wondered what to do next. I started writing emails to my friends, explaining the situation they were in (at the end of the previous session) and idly wondering what they might want to do next. There wasn’t a clear goal in mind here; it was just an excuse to stay in touch and to keep the game going.
Much to my surprise, they all got together and wrote a full-prose reply to my email, with action and dialogue. It was intense and really interesting to read! Given that kind of authority over the narration and the action, the characters and the story developed in the space of just a few pages into something far beyond anything that had happened in our game so far. (That was my first real experience with how empowering players to make creative contributions can really change a game, and I often think back to it.)
My second comment is for those people who choose to keep such things private:
I would never assume that the other players will be interested to read your fan fiction/made-up in-universe stories (you might be disappointed!), but I would always read anything like that someone in my group put together. It’s interesting and it enriches our roleplaying - a win-win. Unless they produce so much material that I literally can’t get through it all, I’m always interested in this kind of thing, and so are most people I know. So, if you’re just feeling shy, go ahead and make it public! So long as people know they’re not required to read it, it should only entertain your friends and improve your game.