I think that perhaps you misunderstood something...
The option to Narrate a Story Element is meant solely to get the player more engaged in the game. The reference to "other players" is only to mean that the story element should not require any rolls to be made by anybody, not that the story element should be about other players.
For example, Joe wants to narrate a story element. In doing so, the bit he narrates should not require Henry to make a skill roll to attempt to do something, nor should Jim need to make a skill roll for it, nor should the GM need to make a skill or saving throw against a skill roll.
And the purpose is to "move the story forward", not to benefit any individual character in any way. And of course, the GM actually has veto power over it (in case it conflicts with something he is planning, or if it is abused in some manner in an attempt to bypass parts where skill rolls are required.