Running Parallel Campaigns
My experience with running parallel campaigns mostly involves the World of Darkness games, Aberrant, and Call of Cthulhu. Some ran longer than others. The WoD games lasted years of real time, and were designed to be open-ended.
I used Vampire: The Masquerade as the baseline game, with players in three separate factions (Camarilla, Anarchs, Sabbat). Added to this game were additional weekly sessions with different players in a Werewolf Chronicle in the same setting, monthly sessions with different players in a Mortals Chronicle, a weekly Mage Chronicle in the same setting, an infrequent Mummy Chronicle, and a small Wraith Chronicle. In Aberrant and Call of Cthulhu, I ran sessions designed to be of a more structured and much shorter duration using the same settings but with players in various locations both in reality and in the game world. This was a method of maximizing preparation time from my D&D days where I had several small, regular groups of players that could not meet at any set time or place to play together.
Problems common to each campaign
The biggest issue overall I had to face was one of keeping the rate of progress between each group consistent. Many elements can be recycled for each group, such as news reports, rumours, signs, portents, and so on so background preparation is less intensive than you might expect. Your main story threads do need to be very distinct to allow your players the freedom to operate without conflict from the storylines of the other group(s).. Geographic separation is often essential for this to work, but vastly different social status or networks can accomplish the same goal.
Cautions for characters in a shared universe
When storylines overlap there are pressures to use PC characters from other groups as a temporary NPC. This can upset some players, so it's something to be discussed beforehand. From the outset, it is best to decide once and for all if the game universes are the same or separate. You have the most freedom and safety if you choose to have them exist separately, but miss out on the very exciting opportunities for cross-over. There are pros and cons to using and not using PCs as NPCs in other groups, having guest appearances by members of the other group during nights of heavy involvement with that NPC, and developing alliances or rivalries. These decisions are not ones that you should make on the fly. Consider them carefully, and consider the personalities, and personality interactions among your players.
Cross-overs: A useful, short-term benefit
Some examples of manageable and useful cross-over between groups from TV are X-Files and Millennium, the various characters from CSI visiting one or more of the other cities, characters from one Star Trek or another mingling with another crew for reasons of age, change of service, transfer of assignment, time travel, etc, and federal law-enforcement working with characters in a state or municipal jurisdiction with locally based characters.
A great moment from my joint Vampire, Werewolf, Mortal, Mage , Mummy, Wraith Chronicle was when characters from three of the five distinct storylines all came together to either support or oppose the return of a rival Vampire Prince. Players went into the session knowing that there might be PC vs PC interaction and that the session would change many things about all the settings for good or ill (from their perspective). It was like a combination of sweeps week and a season finale. Those not comfortable with going up against other players given equally important tasks to do, but away from the 'danger zone.'
Competition: A benefit or a curse
Either in a cross-over or in indirect competition by working on the same puzzles and storylines in games with the same setting, but in a different game universe competition can be a wonderful motivator, or a massive problem. Not everyone likes to compete, and many who choose to roleplay do so because they enjoy games, but do not enjoy being in direct competition with each other. Introducing members of another troupe of gamers operating in the same universe can backfire for out-of-game reasons, or it might light the spark which really gets both of your groups invested in the tale. Communication and observation are the
keys to discovering which you are likely to get.
This style of campaign will require you to set up the world as a sandbox, and be very aware of relationships, the passage of time, and the rate at which information can reasonably spread. The more you develop the world as an open sandbox with its own events, actions and reactions, the easier it will be to manage in the long run.
It is definitely worth trying if you have the interest and the players.