The problem with that is that it contradicts the story of DESTINY. The whole idea behind DESTINY is that humanity narrowly avoided its own extinction, and is now consolidating its position in the galaxy while hostile alien threats start working their way towards earth. The players take on the role of "Guardians of the City", so they are all effectively allied. To have factions springing up and fighting each other will endanger mankind's future, so I doubt it's going to be anything like that.
I'd say it's more likely that any in-fighting among the players is going to arise at the very fringes of the galaxy. Humankind is starting to explore its former colonies, which went into decline during "the Collapse". Nobody seems to know what the Collapse was, so I'm guessing that a large part of the story is going to revolve around what actually happened during the Collapse that led to mankind's decline. There's probably going to be some evidence out there that players find, which opens up plenty of potential for factional fighting: do you make public what you find, and potentially cause panic on earth? Or do you keep it to yourself until you find more evidence to support your theory of what happened, or disprove it?
That way, the fighting would be limited to skirmishes on the edges of the galaxy, but as Bungie start revealing more evidence in future updates or releases, more and more people would start joining factions.
Let's not so readily dismiss Crispy's speculation, because at this point, we really don't know what Bungie plans on doing, and both Crispy and your descriptions/ideas could very well be a direction they could take (Crispy's is less likely, of course). But to say that such a thing goes against the ideals of Destiny assumes that everything that happens is locked into place to a degree that we really don't know. I'd like to think that, after hearing and reading and seeing some of the ideas they're talking about, Crispy's speculation could readily be possible.
But the important part, is that it wouldn't necessarily go against what Destiny is about. The Covenant were trying to expand and establish themselves in the Galaxy and they had a massive shift in alliances through half of their forces which caused a civil war among their soldiers.
But an even better example would have to be the Horus Heresy (from Warhammer 40k, Gamesworkshop's lore-filled table top warfare game). In 40k, a similar situation happens to humans, with the planet earth suddenly being cut off from their vast empire amongst the stars by a "Warp Storm". After the storm cleared, Humanity was lead back out to reclaim their worlds by "The Emperor". The Emperor created 20 genetically altered sons which were suddenly scattered out into the stars that he had to find.
Well he set out with armies that were modeled in each of his son's image and genetic makeup, and upon reclaiming the human worlds (unlike Destiny, humanity was still alive and thriving, but they were cut off from each other), he would occasionally find one of his "Primarch" sons, who had all gone on to do amazing feats and unite the world's populations that they had landed on. As the human Empire re-grew, half of the sons turned on the other half in Heresy, lead by Horus (thus, the Horus Heresy). I apologize if you're already familiar with 40k lore!
Anyways, it's not like something similar couldn't happen with a huge group of determined individuals!
EDIT: Although, for this to all happen, it would be under the assumption that the name of the game is meant to have a different meaning, or that it is used ironically.
