That is going to depend on the details of exactly who does what, and how much is written down. It will also depend on the agreements between the people involved, which may well be verbal or implied. I don't know of any actual cases where such a dispute came to a copyright lawsuit. As copyright suits largely deal with financial harms and damages, and in most individual game there is little or no financial impact, a suit might be unlikely in most cases. But lets look at some possibilities.
In the typical case, the game organizer or DM creates a setting and outline for a series of sessions. This is likely to include maps, descriptions of rooms or locations, descriptions of artifacts, descriptions of non-player characters, locations and types of monsters and other hazards, and other such background. This is probably mostly or fully written down (which includes being recorded on some sort of computer file.) It is therefore protected by copyright, and the initial owner of that copyright is its creator, probably the DM. (In some cases the DM uses a published "module" and makes only limited changes, in which case the DM would only hold copyright on those changes, if anything.)
Then the players create characters, in a process that may be supervised by the DM, and possibly participated in by other players. Each player would probably hold copyright to the description of his or her own character, but the DM or other players might be considered to be co-authors of that, depending on their degree of participation.
Then the players start having playing sessions, each player controlling the actions of a character (or perhaps more than one), and the DM more-or-less controlling the rest of the world, guided by the pre-created maps and descriptions. Some of those will be shared with the players, or modified (partial) versions of them will be. Events of the sessions may or may not be written down (or otherwise preserved in a tangible form). If they are not, there is no copyright on them. If they are, there is a copyright, and all the players and the DM are probably co-authors.
Now lets look and agreements and ownership. The DM probably owns his or her notes, S/he has at least permitted players to use any that have been revealed to the players. The group has probably agreed (implicitly) to co-create the adventure as it happens. There probably are no written agreements about any of this.
If a dispute arises, the DM is legally free to withhold the campaign notes, maps, etc that have not been shared with the players. Whether s/he can legally demand that the players not further use the parts of the setting that had been previously revealed is not so clear. It would depend on what the agreement between the DM and the players said; on, in effect, what license the DM had granted to the players. And since there was probably no written agreement, and quite likely not even an explicit verbal agreement, it would depend on what a court found they had agreed on by implication, based on their actions, and the usual customs of such groups. Sorting that out would be loads of fun. It might involve significant legal fees.
The DM is unlikely to hold any exclusive copyright on the previous events of the campaign. Most likely everyone who participates would be a co-author of those. Unless there is a specific agreement to the contrary, any co-author of any shared work has the right to use, and license the use of, the shared work, although s/he must account to the others for any profits.
The DM certainly has no legal right to forbid the same group from playing together with a different DM (absent a very improbable contract giving such a right). The DM can only control the maps and notes s/he created independently.
If a professional DM is involved, as a comment mentions is possible, it might be more likely that an explicit wri9tten contract between the players and the DM will have been agreed to, If so, its terms will control, whatever they are, unless they are somehow contrary to law. Such a pro DM might be more defensive of copyrights, but would not have any more rights than any DM, except as a specific contract might have provided them.
I doubt that the courts will see many actual lawsuits over such issues.