This needs to be implemented: Everyone is always eager to pick their own favorite songs and sharing songs like this is an awesome way of discovering really good beatmaps/songs that people haven't played before.
However if i could just add a few more suggestions.
Lets take the example of a lobby named "4* to 6* host rotation every 2 songs*
Now, the maker of the game (the host) should have an option to turn on song-selection rotation, as per the original idea, but also the option of putting a force-filter on the difficulty of the song being selected, in this case 4stars-6stars only.
Secondly it'd also be nice to choose after how many songs the song chooser is changed, in this case every 2 songs, so that the option isn't limited to just 1 song each.
Thirdly, it'd be nice to add a feature where players can opt-out of being part of the song-selection rotation, for example, those people who don't have many maps or for whatever reason don't wish to pick a song for the lobby.
Fourthly, and probably most importantly, add some sort of anti-AFK timer, like the song-selecter has 120 seconds to pick a song or else the song-selection goes to the next person.
Some other ideas to play with:
Does the host get to force-start the game, or does the song-selecter get to force start the game?
Should there be a feature added where lobby joiners can vote-kick an afk host or song-selecter?
Should there be a feature to forcefully decline a song, for example, if someone picks Freedom Dive 4D and no one feels like playing such a long song, users can "vote" to forego the song forcing another song to be selected instead (this is probably communicable through chat, but sometimes people just don't listen, cuz they're D bags)
Should there be a feature for the host or song selector to put a timer on when to start? For example, make it so that the game force-starts in "30 seconds" or whatever defined time and announce the timer to the whole lobby. Or with this timer active, if 100% of players in the lobby ready-up, it will automatically start the game.
In either case, some sort of this implementation NEEDS to be done, it's just extremely useful due to how public multiplayer games always tend to be played.