stealth_nsk
Deity
I do not believe city state quests are competitive. I believe in one livestream there was a quest to build an Acropolis, specifically, which only the player, Greece, could do as it is their unique district.
It's possible others see this quest as "Build a Theater Square". Anyway I don't think I've seen a human player being beaten to any quest if it's totally ignored. So probably either each plauer gets his/her own quests, or quests are fulfilled separately, with each civ getting an envoy on completion.
I believe we also didn't see quest like destroying particular Barbarian camp, which can't be completed by only one player.