Let's say a group of friends are playing D&D 5e.
Would it be okay for each dungeon to have a different DM and for each to put their own spin on the story?
Every player and the DM of the round had his own PC, and each rotation the DM's character would be played as an NPC accompanying the party. We did 4 stories with 3 DMs (2 players had no interest in DMing). It was fun; the stories had different flavors at every rotation.
5 of us were playing. Me and 2 others were discussing the idea of a rotation with everyone eventually playing and DMing; the other 2 didn't like DMing, and they were left out of the rotation ("If isn't fun for you, don't do it. No hard feelings."). Everyone made a PC for the party, a typical one indeed (Fighter, Bard, Cleric, Rogue and Sorcerer).
The 3 DMs agreed on different level ranges for their stories; for the sake of surprise we exchanged little info about the transitions, but they went smoothly with just those few pointers (the youngest of us has DMed for 12 years already). One story went from levels 1 to 3, another from 3 to 5, the next from 5 to 9, and the last from 9 to 11. Unfortunately, the real life prevented us and the story from progressing beyond that.
While every DM had his own secrets for their part, we sometimes consulted with each other to bring the best story possible from the characters within the story. One of my friends took something that had happened in the first arc (not his arc) and used it magnificently in the 3rd arc. Even the DM of the first arc was surprised because he never thought of doing that, and was happy about having his player character deal with a consequence of a minor NPC of his.
The Dungeon Master's Guide actually contains rules and suggestions on one way to run it. You can read about them on page 169.
I've pulled out two helpful quotes to illustrate how it works:
Each player starts with 1 plot point ... A player can spend no more than 1 plot point per session ... Once every player at the table has spent a plot point, they each gain 1 plot point.
Then look at Option 3, amusingly titled The Gods Must Be Crazy:
With this approach, there is no permanent DM. Everyone makes a character, and one person starts as the DM and runs the game as normal. That person's character becomes an NPC who can tag along with the group or remain on the sidelines, as the group wishes. At any time, a player can spend a plot point to become the DM. That player's character becomes an NPC, and play continues.
If you read the full page, it gives some suggestions and guidelines for how that could work, and how to run the table.
You could also come up with your own system for multiple DMs (for example, one per session might be easier to handle), but it's definitely an idea supported by the DMG.
This has been a pretty common practice since D&D was invented. Typically, in my experience, all the players (including the DMs) had at least one character, usually several, and each DM had an area on the world for his adventures to occur on. The DMs all agreed on how characters would get from one area to another. Whoever was running that adventure, everyone else's characters would travel there and play it. If someone wanted to run a parallel adventure at the same time, everyone could use another character. This also allowed adventures of various levels to be run; everyone playing just brought one of their characters that was of the appropriate power for the adventure being offered. This setup allowed for maximum flexibility and autonomy for the DMs; little coordination was required.
If you wanted a more continuous campaign, the DMs would have to coordinate a bit to make the transitions smooth. Probably in that case, the PC party would be pretty constant, with each player, including the DMs, having one character in the party. When a person who had a character in the party was DMing, he could either play his own character as an NPC, or entrust it to be a second character for a player he trusted, or have his character stay behind in the city, or engage in some other activity, during that adventure. There are lots of way to handle all of this, but yes, it is entirely possible.