I do both.
My NPC's have their motivations. They have actions they enact during the game while my PC's are doing their things. The "Story" is where those actions intersect.
I merely present those interactions on either side of the equation and give my PC's all the leeway they can muster (probably more) to let them in on the NPC's motivations and their actions which of course influence their play. And I present all of this with as much detail as necessary to raise the fun to the highest level - or to extract the emotional reaction (desired or not) that elevates the game.
The story is simply what emerges out of those interactions.
Greetings!
Yep, Tenbones, I agree. I do the same thing, more or less. Sometimes, the Players really jump in and drive "The Story"--and other times, the various NPC's around them--their friends, henchmen, lovers, family members--also engage in and push their own "stories". Meanwhile, the various villains, other NPC's, factions, tribes and whatever in the wider campaign, they too, often have their own agendas and "stories" that they pursue. The bigger story, of course happens when all of that criss-crosses and intersects in interesting ways.
The Player Characters are often central, of course, and serve as "prime movers"--but they aren't the only story being told, or advanced. The world has its own population, with lots of other heroes, villains, NPC's, that are all each doing their own thing, regardless of what the Player Characters are doing. Naturally, at points of interconnection, such NPC's shall respond to the Player Character's actions and motivations. The entire world doesn't just stand around with their thumb up their asses waiting for the Player Characters to "Get Involved." Sometimes, the Player Characters can act bind, stupid, or otherwise entirely clueless, and can oftentimes suffer the consequences for such dithering. The world must always be active and dynamic.
Semper Fidelis,
SHARK