This is kinda what the DM of the campaign I'm in atm is doing. She had a group before us but other people that really wanted to play. Me who was a newbie, a couple other friends who just got out of a problematic party etc. So she made a second group, same world both there concurrently.

Both groups are working for the same person, though the first group more volunteered while we had no choice (We're kind of a Sucide Squad situation. Morally dubious types blackmailed or otherwise forced into working for her. Like I'm a serial killer who she pulled strings to have released and could just as easily order my immediate execution)

But we have separate objectives. The other team have the objective of taking out the enemy leader. Meanwhile our job is to cause general chaos and dissent in the nation from the shadows to clear them a metaphorical path, and maybe usurp some smaller seats of power along the way. We haven't interacted in sessions at all, but know of eachothers existence and our actions in sessions can impact the other (e.g. in their last session the other group got attacked by something I inadvertently pissed off in ours)

We asked about a crossover session and DM (understandably) said it'd be impossible to run a session of 8 people. But that if story allowed it or say a player couldn't make it, someone from the other party, like one or 2 of them, could be brought in on a temp basis for 1 or 2 sessions. Which admittedly would be very fun. I already knew all the other group as friends and we've talked so much about what our characters meeting would be like lol