If it works anything like PC (which I believe it should), when playing a local co-op game, the world is available only when the player that hosted is playing as his/her machine becomes the server of that world. So, for example, you start a local game and invite your friends (or they join you). They can only access that world when you are actually playing that local game.
The good side to this is that you and your friends can each have your own local games that each can play as single player when the others are not available, but also try different play styles and builds on each others local games.
Now, an official PVE Server is always available to all of you (and any other player that chooses it).
Private (rented) servers are always up unless the admin (player who rents and controls the server) takes them down for any reason such as adding/changing mods or server settings or periodic (and usually very quick) restarts to improve server performance, decrease lag, reduce/remove glitchy behaviors, etc. Or the rare occurrence of the server company shutting down servers temporarily due to issues or emergency maintenance and such.
I love the idea of co-op games, but when trying it last summer, the rubber banding (tethering) was so terrible, your friends had to always be right on top of you. More recently, they’ve added settings for the tethering that they seemed pretty happy about in one of the developer streams, so it might be a great option now.
I used to run a private role play server with mods on PC last year and will again after launch, the developer’s kit update and excellent mods such as Pippi are updated and working. Renting a server is a nice solution though it can get costly. Set up the server settings and/or mods and play the game your way with your friends (and others if you choose) and no need to worry about the world disappearing if you have to log out.
Of course, you and your friends could also explore the official Funcom and/or other private servers. You never know. You may find one or some quite to your liking. I found a few private role play and “builders paradise” type PVE servers last summer with great communities.