I generally have no problems with the players influencing things. Since I was playing D&D, there are no real mechanics for players directly narrating, but, I would certainly allow most suggestions. If the players wanted to find something, for example, they generally would. Then again, I have very good players, so, their suggestions automatically take the game into account and are intended to improve the game.
A minor example might be one of my players who wanted a magic item with magic pockets. He just wanted something that he could reach into at various points in time and pull out something strange. So, he got it and I used a rather massive random table to find out what he had in his pocket each time he reached in. Sometimes it would be incorporated into the game, sometimes not.