My personal preference both as a player and as a GM, is that this sort of meat-level fiction generation by players happens mostly before the campaign in the PC background creation phase, end when elaboration is needed, between the sessions. During the play, I prefer players to be mainly in immersed character stance and contribute mainly via the actions and choices of their characters. I can play in other ways, and it can even be refreshing from time to time, but I know what is my favourite.
At least to me having the authority to decide fiction outside the causal control of my character puts me from the actor stance to the author stance, and this to me is less immersive. It is fine if this is done outside the actual game sessions, as then we're not immersing into our characters in the first place.
And whilst I like players being able to resolve situations in different ways, I still feels this too works best when there is some sort of fixed reality they need to creatively exploit. Giving the players control of the external reality means that optimal gameplay becomes trying to invent BS that lets you use your best skills. That is something I am good at, but not something I particularly enjoy. And whilst things like letting player to use flashback diplomacy to learn magical secrets might seem fine at glance, it also means that you have now rendered all knowledge skills obsolete.