Quest progression is still done on an individual level. Such that when you load into a world for the first time, you do not have any quests completed.
Quests that are for killing bosses through the Boss Domains however are tied to the world data and killing the boss quest step can be skipped if desired.
Quest steps leading up to the entering the boss domain and killing the boss must still be completed by each player. This is to ensure any “tutorial” related quest steps or information quest steps to teach the game remain and naturally learning occurs.
Upon reaching the “Enter Domain” step of any Boss Domain Quest you will be presented with a “Skip” button on that quest step. Clicking this will complete the quest and reward you with the reward without entering or killing the boss.
If a boss has not yet been killed on the world you are playing on, you may not skip this quest. The skip button only appears when that boss has been marked as downed in that world.
A few examples of what this looks like