The 4e D&D rulebooks have quite a bit of advocacy for the players making decisions about the shared fiction: backstory, the focus of the action, what is possible here-and-now in terms of action declaration.
Some examples around the focus of the action:
PHB p 258: "You can also, with your DM’s approval, create a quest for your character. Such a quest can tie into your character’s background. For instance, perhaps your mother is the person whose remains lie in the Fortress of the Iron Ring. Quests can also relate to individual goals, such as a ranger searching for a magic bow to wield. Individual quests give you a stake in a campaign’s unfolding story and give your DM ingredients to help develop that story."
DMG p 103: "You should allow and even encourage players to come up with their own quests that are tied to their individual goals or specific circumstances in the adventure. Evaluate the proposed quest and assign it a level. Remember to say yes as often as possible!"
Players come up with quests for their characters often in my games, regardless of the edition. It's not typically very formal, and I don't need to assign levels or anything - the player decides their character has a goal, and pursues it via deciding to take those actions in play. They decide where to go and what goals to pursue.
I do let players know that they still need to have characters that are on board for whatever the group is doing, so we don't pull in too many narrative directions at once. And, not every player feels great leaning into this kind of roleplaying. Some prefer to be a little more reactive - game for whatever the current narrative is.
An example of that is my gothic fantasy snow white campaign. There's a reason the party is called together, and it defines the arc of the campaign ("steal the crown of the princess of the cursed kingdom for the rat king.") Why the players pursue such a goal is something I explicitly put in their hands, and some are a little reactive, and others are a little more pro-active. I've got one player leaning into the idea of restoring the order of Looking-Glass Knights that once defended the queen, so I'm putting in effort into allowing him to do that through the narrative (there's an NPC he can specifically inspire and forge common cause with). Another is explicitly learning how to become a Thief of Stories, so that element gets a bit more work (through another allied NPC). The party Fighter, though, is just kind of along for the ride, and is having fun being along for the ride.
What I find useful for encouraging this is a light touch with the sort of intended campaign arc. Often, I'll even write the arc
after the PC's make their party, so I can specifically highlight classes, races, backgrounds, etc. Makes envisioning the path easier, anyway.
Some examples around what is possible in action declaration:
DMG p 42: "Your presence as the Dungeon Master is what makes D&D such a great game. You make it possible for the players to try anything they can imagine. That means it’s your job to resolve unusual actions when the players try them. . . . [rules and guidelines follow, and then an example action declaration] . . . This sort of action is exactly the kind of thinking you want to encourage . . ."
DMG pp 73-5: "When a player’s turn comes up in a skill challenge, let that player’s character use any skill the player wants. As long as the player or you can come up with a way to let this secondary skill play a part in the challenge, go for it. . . . players can and will come up with ways to use skills you do not expect. . . . Characters might have access to utility powers or rituals that can help them. These might allow special uses of skills, perhaps with a bonus. Rituals in particular might grant an automatic success or remove failures from the running total. . . . Thinking players are engaged players. In skill challenges, players will come up with uses for skills that you didn’t expect to play a role. Try not to say no."
Yeah, happens all the time. I encourage a "what would your CHARACTER" do mindset to help get away from the menu of action choices and get more into the land of character goals.
Last session, a PC dragged an exploding lightning-ice-golem-thing away from the rest of the party. I've got a halfling often perched on the back of an orc.
I don't really need a specific construct like a skill challenge to encourage this. Personally, the skill challenge structure always struck me as over-designed and gamist, so I typically rely more on a call-and-response kind of format where I just ask what the PC wants to do, and then tell them what skill to roll (often using group checks when they all want to do something similar).
Some examples around backstory:
DMG2 p 12: "A sense of shared authorship between you and the players can begin before you start playing, when you create the campaign. . . . Have each player bring a pitch, a basic idea for the campaign . . . The pitch is a simple sentence that describes how the player characters fit into their world."
DMG2 p 15: "You might also ask players to invent one or two NPCs to whom they have important ties. These can be ties of loyalty . . . Alternatively, these NPCs might despise the character, and you can use the NPCs as obstacles to the character's goals."
DMG2 pp 16-17: "The process of shared creation doesn't need to stop during your campaign's prep phase. You can continue it by allowing players a role in inventing your D&D world. . . . When you are presented with player input into your world, start by repeating to yourself the first rule of improvisation: Never negate. . . . Three main techniques allow you to bring player suggestions to the fore: incidental reference solicited input, and the turnaround. Proactive players might employ a fourth method, the direct assertion."
I don't formally collect player input or suggestions for world elements, but I do work player elements into planning as I described above, and I do get their buy-in for the big idea of a campaign. In my current game, the characters were strangers to the world, and so presenting them a land that they did not know and had to figure out was part of the appeal. Having them design an NPC would've been at cross-purposes with that appeal. In more open-ended games, I like to have the PC's tell me what they're interested in at character creation, and then build the slice of the world they'll see through the lens of their characters. Like, a PC choosing the Criminal background might mean there's a criminal organization that'll be prominent in the game, and a PC choosing the Shadow Sorcerer subclass probably means I'll be referencing the Shadowfell at some point, and a PC choosing to be a dragonborn probably means there will be dragons, and if that is all in the same PC, maybe I have a shadow dragon who sits at the core of a network of thieves as one of the villains of the game.
When I used to GM Rolemaster and AD&D, players would contribute around backstory, and the focus of the action, but not so much in terms of what is possible in action declaration. Those systems don't support that sort of player contribution like 4e D&D does!
Who else's D&D has a high volume of player-generated fiction?
I don't personally find that the edition matters that much for me - this is how I approached AD&D and how I approach 5e. When I ran 4e, I found a few elements worked against this style, but not enough to invalidate it. Like, the formalism of the skill challenge was never anything I adopted, and I didn't really miss it, and I don't think I had fewer moments of improvised skill checks because of that. Folks still told me what their characters did and I gave them a skill to roll.
