People say all kinds of stuff during play. The question is who gets to say what so that it sticks. Meaning, we're going to proceed from this point having accepted what was said into what's going on.
When you cut the goblin's head off, do you look at the dice and whatever, and say so? Or does it not quite count until the GM says so?
Four distinct types
Content authority - over what we're calling back-story, e.g. whether Sam is a KGB mole, or which NPC is boinking whom. This includes preparing such information at any point as well as revealing it in play.
Plot authority - over crux-points in the knowledge base at the table - now is the time for a revelation! - typically, revealing content, although notice it can apply to player-characters' material as well as GM material - and look out, because within this authority lies the remarkable pitfall of wanting (for instances) revelations and reactions to apply precisely to players as they do to characters.
Situational authority - over who's there, what's going on - scene framing would be the most relevant and obvious technique-example, or phrases like "That's when I show up!" from a player.
Narrational authority - saying how it happens, what happens - I'm suggesting here that this is best understood as a feature of resolution (including the entirety of IIEE), and not to mistake it for describing what the castle looks like, for instance; I also suggest it's far more shared in application than most role-players realize.
Authority is a Techniques issue which is all too often handled (or not handled) as if it were a Social Contract issue.
Historically, this squished-together mass of Authorities called "the GM" is set with one person throughout play, axiomatically. This fixed assignment can take on serious Social Contract content and become a virtual identity and real-world authority issue, including being confounded with Leadership.
More functionally, a given group may assign the person who typically takes on the GM role its own profile of authorities, which is fine until someone joins the group with a different idea about what that profile is supposed to look like. This gives rise to some of the trouble with the concept of The Good GM.
Distributed vs. centralized
Historically, role-playing culture assigns one specific person most or all of all four types of Authority. Such an arrangement is viable, but the assumption that the four types must be held by one person is not.
Parceling out the different authorities to different people is surprisingly functional. This turns out to be a perfectly viable rules feature
Fixed vs. changing
Rules and practices may also include re-distributing one or more authorities at different points during play. Basic examples include shifting the full GM role to different person between plot-based units of play.
Examples of dynamic and distributed authorities in game design
The poster child game for distributing narration authority mechanically is The Pool, including a specific option to decide between narration authority and gaining a unit of Resource at the moment of a successful roll. The game which re-distributes all four authorities as a formal feature of turn structure is Universalis, which also permits bidding duels to transfer authority at key moments.
"If someone has Authority over one of the above types, then he or she must necessarily have it for all four, or else play will fall into chaos." This is a deeply cherished concept in role-playing, but it's wrong. For example, sharing Narrational Authority by itself poses no threat at all to centralized Content Authority. Also, what actually happens in a given group may not be what they say or think happens - Situational Authority, for example, may be nominally held by one person, but in practice, people frequently request scenes for their characters, or state a character's actions and expect a corresponding scene to be held for them.