It's great to involve the player in an epic struggle through faction conflict, such as the Galactic Civil War between the Empire and the Rebellion in Star Wars: Galaxies. But that epic conflict doesn't have to be the only faction conflict, and players don't have to owe allegiance to only one community.
All persons have multiple allegiances. You might have loyalties to friends, to family, to your nation, to your local area, to a club, to a religious or political organization, and on and on. So if it's so common to have multiple loyalties and participate in multiple faction conflicts in real life, a similar situation certainly wouldn't feel strange in a game.
One way this can show up is conflicting loyalties, a dominant theme in most successful stories of fiction. If you played Deus Ex, then you might remember your character having to decide between his loyalty to his government employer and his loyalty to his brother (who the government agency claimed to be a traitor), with a strong consequence in terms of gameplay. Likewise, an MMO player might have to decide between a larger faction and a more local faction.
But it's also possible to allow a player to join multiple faction conflicts which do not directly relate to one another. The player might be a soldier in the epic war (pick a side), a crafter in a city's Traders Guild (which competes with other cities), a frequent military escort for a friendly local NPC (who the local bandit faction likes to target), a defender of the region's "Hadrian's Wall" (which holds back some really nasty monsters only so long as players continue to cull their population), and so forth.
Characters of any fiction are often more interesting when they have multiple loyalties, multiple goals and considerations; when they are not just stick-figures that look the same from every side. Aside from offering players a greater variety of gameplay options, allowing players to join multiple factions and on-going conflicts would make their characters deeper and more worthy of attachment.