Both.
I think metaplot on a game should come from Staff. I think Staff should run plots. I think the majority of a game's staff should be Storytellers who keep the game going by helping tell the game's story.
I think player-run plots are important for stories that impact only a few. For those personal stories that really grind your character down. Player Storytellers should be the ones helping tell the character's story.
The overrealiance on players to run things (e.g. The Reach's Tier system) is, to me, part of what makes up for so many people's frustrations. Staff should be in the thick of things. There should be NPCs that, while not characters in the same way PCs are, can be interacted with and weigh on the game world as much s the characters, or at least almost as much.
Vampire is a political game about undead beings both young and ancient playing deadly, cutthroat games of intrigue. A game with vampire should foster that feeling, that ambience, that sort of setting through both its location (easy enough) and its NPCs (not so easy). Have the Praxis be a(n) (un)living thing. Use the second edition's Aspirations system to give each NPC in the Praxis a Long-Term Aspiration, and then give players surprise bonuses or even surprise experience when they help those NPCs achieve those Aspirations, even if they were manipulated into it, even if they had no idea it was going to happen. If a neonate PC talks back to the Prince, even if that Prince if an NPC, make sure there are consequences: string them up, Torpor that uppity young kid and then torture them in their Torporic dreams. and if they do it again, call the Blood Hunt. That's the game. Just because the Prince is an NPC doesn't mean characters should be able to get away with shit that they shouldn't be able to. Play the game that you made a character for.
That's a very long tangent and I think it's clear where I would be going if I moved on to examples for the other games.
TL;DR: Staff plots should be the game's story, and player plots should be the characters's stories. There is room for the two to switch and even to mix, but that should, IMO, be the default. While I don't begrudge anyone making a game with no metaplot, so to speak (e.g. Reno), I don't think it's the kind of game that can keep its momentum.