Machiara:
I don't think it's a fallacy, because I have personally seen hidden threads damage game communities.
The fallacy is the assertion that hidden threads will
necessarily harm the sense of community. In some games, with some players, separate threads
can harm the sense of community, but as in most things in life, role-playing is a deeply context-dependent activity.
I've done it both ways. And had success (and failure) in both cases. As many have said before me, it depends on the game, the players, the story, the groove of what it is you're trying to accomplish. I'm rarely worried over-much about metagaming. I've found that a little bit of metagaming can be an important part of any story. But not everyone feels that way about it.
Whereas I have a deep, deep hate on for even casual godmodding ("Bill walks into the bar and the bartender is shocked at the wounds on his face.") Even a little of that sets my teeth on edge. But not everyone feels that way about it. One of my regular players is prone to it, just a little, and I've learned to let it go because I like playing with them so much.
But in regards to separate threads, for example, for a game I'm running now, the structure of the story is that the players will come together as strangers to deal with a common threat. And after that, they will part for several years before being forced back together again.
Because I want them to start as strangers, I'm doing the initial workshopping in Private. And because they'll be separated, I'll be doing the inter-adventure part in private too. The only information I'll share with everyone are exploits that 'everyone' would have heard about through gossip, legend, etc.
So when they come back together again, the characters and players will have imperfect knowledge of what they've all been up to in the interim.
Some GM's are really big on character knowledge being clearly separate from player knowledge (sometimes even to the point of not allowing players to post internal dialogue). Which is fine. That's their groove. That's how they have fun. For those GM's, separate threads would be vital to maintaining the mood and feel of the story they want to tell.
tl;dr - It totally depends on what kind of game you're running and what kind of story you want to tell.