To state the obvious, the changes in how to run a DM+1 game as opposed to the traditional approach come from having only one player. I'll talk about how to approach the narrative and mechanics parts of that in future posts of this series, but I want to talk first about the interpersonal impacts, specifically as they relate to attention, collaboration, and pacing.
When people outside of the TTRPG hobby ask "what is D&D?", it's common to draw comparisons to movies, books, or video games. Putting aside the mismatch of linearity and narrative limitations (as discussed here by Justin Alexander), another common thread in those examples that runs contrary to traditional D&D is their focus on a single protagonist. Yes, there are exceptions, but for every Lord of the Rings, there are several more like Conan the Barbarian, Elric, Die Hard, Indiana Jones, or Star Wars. Even examples where there is a primary protagonist team often have a particular character who gets the real focus (e.g. the Captain in each Star Trek series, Rand [and Perrin and Mat, to a substantially lesser degree] in the Wheel of Time, or Croaker in The Black Company). This is entirely natural, since it's far easier for a member of the audience to relate to a single character than to a whole group.
Furthermore, from my experiences seeing complaints about sharing spotlight time in TTRPGs, the vast majority are framed as either "nobody paid attention to my character" or "this player kept pushing their character into the spotlight". On the other hand, it's pretty rare to see complaints about the player's own character getting too much spotlight time systematically; at most, they might bemoan having been put on the spot in a particular situation.
A DM+1 set-up addresses both of those points inherently. Not only is there only a single protagonist (thus hewing more closely to the inspirational materials, whether from 1974 or today), but the player controlling that PC is essentially guaranteed to have the GM's full attention whenever they're doing something, which is all of the time aside from when the GM is speaking.
This can have drawbacks, too; not all players want so much attention and the accompanying responsibility. Increasing comfort with the DM+1 style over time can help to alleviate some of the former, as can various techniques for imposing separation between player and character (e.g. speaking in third person about what their PC does, summarizing talking points instead of acting out dialogue, etc.). I also find that players tend to ease into it when I ramp up my own theatrics (i.e. when I show willingness to make a fool of myself).
The topic of responsibility ties in with providing elements of collaboration despite having only a single player. There's no question that part of the fun of D&D is playing with a team (the fellowship aesthetic), and arriving at decisions as a group also shields anyone from bearing the full brunt of responsibility if it ends up being a poor decision. My way of addressing this is to provide more NPC interaction to provide that collaboration without making choices for the player.
For instance, on both this past Sunday and Thursday, I ran DM+1 sessions, and each of them happened to feature the PC exploring a dungeon with a single NPC ally. In both games, the player asked the NPCs for advice about what to do, and I never answered such questions directly (perhaps in part from past experience with exactly this situation, I made sure the NPCs had legitimate reasons to defer). Instead, I had the NPCs reply by voicing their fears, misgivings, cowardice, and/or ambivalence as questions or prompts back to the PC (similar to Socratic dialogue). In that way, I engaged the player in a conversation, provided them with alternative points of view, and pointed out potential flaws in their ideas, but the actual actionable decisions were still up to them. This can even be used to provide information that the player hadn't thought of (e.g. when one player remarked about wishing she could interrogate someone for more information about the tomb she was exploring, her NPC ally mentioned that they might be able to find someone who could use magic to speak with the dead, and she came up with the idea asking an anthropological historian to do that without any further input from me).
This approach allows the player to retain the sense of teamwork without giving them an opening to let the GM take over what happens, similar to what Matt Colville describes in this video. At the same time, while it doesn't relieve the player from being responsible for their choices, it does provide some safety net to see that there are NPCs they can trust to try catching critical holes in their ideas or to offer suggestions about how to bridge the gap between having a near-term goal and figuring out how to accomplish it.
Moving on to pacing, I believe one of the biggest adjustments for the GM is getting used to not only how much faster the game will run but also how much less time they'll have to sift through their own notes or reference materials. Both of these are a consequence of boiling the game down to a constant conversation between the GM and a single player, which eliminates any discussion or deliberation between the players. In my experience, the exact same content takes about twice as long to run for three players as it does for DM+1.
The biggest implication that comes out from this is that you have to know your setting well enough to improvise. That might sound like an obvious thing a GM should do for any game, but it's especially important for DM+1 games because anything that goes beyond what you've memorized brings three sources of pressure simultaneously: nobody will talk the player out of odd choices without your interference, there won't be any reactions from other players to buy you a crucial extra few seconds to think, and the game will be on hold until you can think of a way to resolve the matter. There is a minor out in taking a moment to flip through reference material for a detail, but it had better be something that you can find quickly, or else the cost in lost time will likely outweigh the value of whatever you were looking for.
Now, does this mean that you should be preparing more content for a DM+1 session than for a larger group? Well, that depends on context, which I'll get into more next time when I talk about differences in scope.
Returning to the topic of pacing, another nice thing about DM+1 is that the GM has only one person to watch when it comes to "reading the room". Speed the action along or take a moment to pause and breathe based on how the player is reacting (circumstances permitting, of course). An extension of this is also minimizing, if not eliminating, repeated rolls for the same task. That's good practice in general, but it's especially more important when, for instance, those three Open Doors rolls are being done by one player instead of spread across multiple players, changing it from an impromptu contest to see who's lucky into a tedious exercise in rolling until the player hits an arbitrary threshold to proceed. Failing forward, introducing new complications, and/or succeeding with a cost are generally better than simple stonewalling.
There are cases when a firm failure is fine (for instance, if that Open Doors check would trigger a random encounter check on a failure), but try to be mindful of finding a way to move things on if the cost of failure isn't interesting. Again, that's good practice in general, but it becomes more important in a DM+1 game because most players' natural reaction to a failed roll will be to keep on trying until they succeed, and it's easy for them to get stuck in that rut subconsciously without any other players to jar their mind out of it. Perhaps just as importantly, because the spotlight is focused solely on their PC, it's easy for in-game failures to lead to real-world frustration because there isn't anyone else to provide other successes or at least spread around the failures. There's a post on Delta's D&D Hotspot about how something repeating three times causes the same mental load as repeating infinite times, which I'd guess is also in play here. Two or three consecutive failures can cause the player to grow very frustrated if they aren't making progress, especially since their failures are likely to happen in a smaller window of time due to what I've mentioned above. Granting some measure of success on a failed roll, or ruling an automatic success based on their skill as a player, helps to alleviate that effect.
Obviously, there's a lot more to be said about each of these topics and many more besides. However, DM+1 games are a highly personal experience, and going into specifics about running them is difficult to do without knowing your (and your player's) strengths, weaknesses, interests, and aversions. Thus, this and the future posts in this series will tend to present high-level advice that I've found broadly applicable. Figuring out how to apply it to your individual playing experiences will be left largely up to you, though of course I'd be happy to discuss more in the comments or via email.