Current published answer
The official FAQ says dedicated and private servers are not supported at this time. Co-op uses a player-hosted peer-to-peer model instead.
What that means in practice
A group should not assume a rented server will keep its shared world online independently of the host. Organize persistent activity around the person hosting that world.
Verify after patches
Server plans can change during development. Check current official release notes and the game's multiplayer options before treating this status as permanent.