RPGs of the mainstream sort - which all the ones I play are - have a GM participant, who is responsible for coordinating the backstory/setting and who is responsible for presenting situation.
The key question therefore becomes,
where does the content of situation come from? Someone has to provide it, and that person must either be in the GM role or the player role.
It's important to see that
the player can provide the content even though it is the GM who is actually doing the presenting/framing. For instance, in my most recent Torchbearer session I described the Elven Ranger seeing the light of a comet shining up through a hole in the floor of a stone structure that reached out into the river the PCs were walking beside, the comet's light being reflected off the water and somehow focused or amplified by the structure. The stone structure had been built by Dwarves, and had an inscription in Elvish.
The comet was something that the Ranger's player had introduced into the fiction several sessions ago - after missing a session, the player had to explain where his PC had been (this is a standard procedure for Torchbearer) and (given that the PC has Stars-wise; and given that the PCs had recently experienced a magical catastrophe occurring in the village where they were staying) he told us that his PC had been following a comet as it travelled through the sky, looking for answers to the cause of the catastrophe.
One of the PCs is a Dwarf, and has as his Creed that Elves need grounding in reality. The third PC is also an Elf, who has as her Creed that in these dark times, all Elves need help.
The purpose of the structure's magic was to allow purification of the river water, which flows out of the Troll Fens. The Troll Fens, and their polluted water, had already been a feature of play over the past several sessions, which had focused around the Fens, the Moathouse in the Fens, and the PCs travelling through the Fens by boat and by trudging.
This is different from "story hooks" which are GM authored without reference to player-established priorities for their PCs.
I wouldn't normally describe the RPGing I do as a sandbox. I see "sandbox" as describing a particular technique for framing situations. Here are three such techniques (which are distinct from the question of
content I talked about just above, although in practice they might tend to lean one or the other way):
1. The GM has a map and key that is detailed enough that players, by their declared actions - generally movement-related ones like "I open the door" or "I enter the clearing" or "I turn the corner" or "I go into the cave" - trigger pre-prepared situations. Gygax assumes this sort of process for establishing situation in his AD&D rulebooks. This is what I would call a sandbox.
2. The GM presents "hooks" which may be low-stakes (a quest-giver approaches the PCs) or high-stakes (the PCs encounter something exciting in media res), and which the players - knowing how the game is meant to work - recognise as situation for their PCs to engage in. I think this is pretty mainstream. It does not rely on map-and-key, but rather the GM's ideas about what "story"/"adventure" they want to present to the players.
3. The GM presents the players with a situation that demands a choice or some action from their PCs. This can resemble (2), but the contrast is this: in (2) the PCs (say) see someone being mugged; whereas in (3) the PCs are themselves assaulted by muggers.
I generally favour approach (3), although my current game of choice - Torchbearer 2e - uses (1) as well on a small scale ("dungeons").
The changes are decided by the GM. The circumstances "allowing for such" is decided by the GM.
As per what I've written just above, the key question is
who is deciding what happens next?
If that is the GM, rather than the players, then what I see is a railroad. I mean, the players might be generating prompts into the GM's decision to say this rather than that, but it is the GM who is deciding.
A non-railroad sandbox requires that the players can
know what the consequences will be. If consequences are following from circumstances, then players must therefore be able to know what the circumstances are. This is why Gygax, in his advice on how to do skilled dungeon play, emphasises the importance of the players collecting information about the dungeon - in practice, by declaring low-stakes actions around moving and listening and the like, so as to learn what is in the dungeon, enabling them to then make informed plans about how to loot it.
A sandbox in which the key information is - in practice - largely inaccessible is not going to permit the players to take control in this way.
And this is precisely the sort of thing that I have in mind - the player doesn't know the circumstances or the stakes, and is in effect declaring actions "blindly" and just waiting to find out what consequence the GM decides will follow.