Worlds are only saved to the host player's console. No one else can load the save. No one else can continue playing without the original host first loading the save. If the host quits then everyone else gets kicked.
If you and a friend are playing a random game and do not know the person hosting it you will have no control over when that game will be loaded and available to play again.
There is also currently a bug that will sometimes corrupt a world save. A patch is due the end of July/ beginning of August to address this issue.