One of the best campaigns that I've ever played in was the first long term campaign I played in, back in my first year of high school (1980). It was kind of AD&D*, run by two DMs and a horde of players. We'd invade someone's house for about 24 hours each weekend. The two DMs were busy nearly the entire time. While all the players were hanging out, you could organize a group to put down the atari, pause the Monty Python and go after something in the game world - and the DMs would run it. They didn't, to my knowledge, use modules but their own world, city and dungeons. All of it was accessible for us to explore, though a lot of it was dangerous.
That kind of player freedom was incredible. Throughout the day and into the evening, you'd hear bits of what other players had accomplished. "Jeff and Chris went off to the island to see the archdruid." Wow - what did they do that for? What did the druid tell them, or give them? "Gabe found a secret door to another part of the 3rd level, and there were puzzles that used the pieces we found on the second." There was bragging of accomplishments (in good fun, we didn't really trash talk much.)
I remember one such expedition: just my character and one other, both elves. We purposefully limited our group to go down into the dungeon without light sources... as well as to get "more XP" for ourselves.
Here's another example: We knew that there were other parties of adventurers going into the sewers after an evil cult, as it was talked about in taverns. But we didn't know how to find the entrances. I got two players together who's characters were thieves, and through their guild they found out which of their fences were buying up items from the cult in the sewers. The three of us then cased the fences, found the npc adventuring party and followed them to the hidden entrance. I think we might have ambushed them too - not every character was good. (In fact the mix of alignments made things interesting.)
The DMs had laid the hook out to all of us but otherwise didn't help us in any way to find it. We were just the first players that invented a way to work the hook. Eventually we let other players in on the secret as there were parts we weren't tough enough to take on ourselves. We debated that - we wanted the loot all to ourselves. But the risk of death (it happened) was too great. We wanted to invite a cleric, but he wouldn't come along unless others did too.
As you can tell, it wasn't just the NPCs and monsters that were scheming - the players/PCs were schemers, too. Those kinds of player schemes, which depend upon DM flexibility to player initiative, DM willingness to play the whole world rather than just the established dungeon/adventure, are what sandbox play is all about.
Some of that mystery is lost now that I've had so many gaming experiences. But that "where do you want to go, the whole world is available" freedom made exploration and discovery a part of the game, a reward in and of itself, regardless of the XP or gold found. Gabe found that secret door to the other part of the 3rd level, and in a way that level would always be his.
So, as a player in my current game, Zeyya is very possessive of her map to the kobold lair. Ha! As a DM, I record who knows how to get to the cave of the plant people, which of the nearly 20 characters have a relationship with Godfrey, the village headsman of Barleyfield, and so on.