Nearly every server is different, but the ones my friends/wife and I always did (10+ years ago) were like role-playing kingdom building maps. Server owner (usually me) would hold the title of King/Sovereign and appoint their friends to specific roles. I would oversee the general development and expansion of the kingdom, as well as decide and manage a system of ore-based currency (or would at least create the mint and appoint someone to running it). Afterward I would introduce and gradually roll out phases of a larger storyline for anyone who cares.

My left and right hand would build/manage the keeps/barracks/military structures, or the government buildings/libraries/cultural centers, etc. These would all be injected with their own lore and staffed by the person in charge of them. Everyone else would receive more minor roles, but typically be given monopolies in certain types of goods or commerce. Maybe Bob wants to be a trapper. Sure, anyone else can legally go and gather leathers and animal parts, but Bob is the only one permitted to sell those items in his shop in the city. Things like that just to try to keep it interesting. When Bob isn't trapping or trading or being involved with the kingdom, he's pretty much just playing Minecraft on his homestead.

The idea is to open it up to the public (via applications and careful vetting) and watch people run amock in the simulated medieval economy. We used to have a blast doing it. Especially with mods installed that added skill progression, abilities at milestones and other MMORPG-esque mechanics.

Normal people, however.. They just do what they do in single player but occasionally trade, work together, tackle bosses, and show each other their latest creations.