I've seen this sort of thing pulled off, and it was fun.
However, I would warn against PvP. That's just not likely to end well. Regardless of which team loses, the players lose. An especially unlucky player might lose both characters. It's a lose-lose scenario.
Frankly, if you want to save yourself headaches, I'd set the campaign up to have as little contact between the parties as possible. This is in no small part because, otherwise, you'll need to keep exacting records of who did what when. Also, assuming that these groups adventure in parallel to one another, what happens if there's a paradox?
For example, two parties (A & B) start the game on January 1st. Party A plays first, and they get into a fun adventure which concludes in them getting lost. Exhausted, they find their way to a ferry on the 14th of January. He gets them across the river and they return to town triumphant. End of session.
Then party B plays. They decide to try their hand at banditry, and on the 5th of January, they rob and murder the ferryman, and burn the ferry to the ground.
Only, hang on a second. That's a paradox. How did party A get back to town with the ferryman's help if he'd been dead for 9 days, and the ferry was burned down?
In the campaigns where I've seen this most successfully executed there was always something preventing this from happening.
In one campaign, we were playing the heroes in the real world, and the villains in another dimension (kind of like the Upside Down from Stranger Things).
In another, we had been on an important adventure to restore the world after it ended. Our characters decided to recruit a second party that they would alternate adventures with, so that if one party fell there would still be someone left. (Actually it was just because we were leveling up a lot faster than the DM expected.) In that campaign there was occasional cross-over, but it was always intentional. One time when one of the parties TPK'd and the other party went in to find out what happened to them (and recover their corpses). The other was when both parties teamed up to slay an extremely dangerous pseudo-deity.
That campaign also introduced a third party on the surface of the planet (the other two parties were at the heart of the planet). This party didn't even meet the other party until the final epic battle, where they showed up as reinforcements, albeit a little too late to contribute meaningfully.
My point isn't to dissuade you from proceeding, but rather to urge you to carefully consider how to handle these issues.