Okay, cool. So then what role would you say your Prep plays in an investigative scenario as opposed to a more sandbox style that you've mostly been discussing? Is there the same "back and forth" between GM and Players as you've described in a Sandbox? Is it more one way? Something else?
Prep plays a very different role in investigation. In an investigation adventure I know the players are going to have sites of investigation, that there will likely be a hook presented to them in a very clear form, etc. There is a clear scenario: investigate the disappearance of so and so, follow a trail of clues, etc. I am going to be prepping the trail of clues, the events that led up to the mystery, etc. I may add in some living elements (i.e. maybe The Red Parrot has a lot of flexibility to move around and thwart the pcs efforts if he catches on to them for example), but it isn't quite like a sandbox where I am leaving the 'what will the players be doing tonight' blank. In a sandbox, I am building them an environment, with pieces that are 'alive', with active 'living' organizations, in a world with ongoing conflicts, with lore and locations that can be explored, and allowing the players to do what they want (and I will happily build on where they choose to go or focus on during play). The difference is pretty significant in my mind. So much so that I definitely prep in very different ways. In some respects an investigation is more labor intensive because I have to flesh out the clues, etc. But it is also more focused, because I don't have to worry about prepping the next town over. In a sandbox I need to prep enough that the players will have stuff to do no matter where they go, but also so that those places will be responsive and be able to take on a life of their own once the players start steering in clear directions. A sandbox may include things like dungeon delves (which I think people here don't need any explanation of how to prep), but overlaid on top of that is the living setting. It makes it more likely to have situations arise where rivals show up during their delves to beat to them to the manual. I think sandboxes, ultimately, become about the characters.
In terms of mystery investigations and sandboxes, neither is back and forth between players and GM in terms of prep. You prep the mystery, you prep the sandbox. Some GMs I am sure take input from players before hand. I usually don't. When I have that has been the exception (and it made for a very different kind of game). The back and forth I was talking about was during play (and as an edge case during things like character creation when players are establishing stuff like family details: though there are sandbox approaches that don't allow for that).
During play though, the back and forth is very important. Ultimately what I want is to have to do as little as possible to keep this thing running. I want the players coming up with stuff they want to do, finding things in the setting that interest them and taking those in wild directions. Some of the examples Justin Alexander gave towards the end of his video are the sorts of thing I have in mind here. The classic example for me is the players deciding they want to form their own sect, or deciding they want to seek out particular masters and become their students, then go forth and make a mark for themselves. When players are trying to establish themselves in the martial world, something they may start doing is looking for opponents to defeat so they can grow their reputation. This will often be knowledge dependent and players may ask me (often by using their skills, or just asking me if they know about something---sometimes going through NPCs to get the info) "Is there a guy around here who is known for strong leg techniques, I want to challenge someone like that with my own kicking style to prove mine is better". Maybe I have a person like that in the setting I've established. But if I don't, I might simply decide 'yes there is' or 'no there isn't' based on what I think is likely (and if I am not sure I may just leave it to chance and roll a die). Then I would rapidly create that character on the fly, and establish concrete details in my notes (personality, name, techniques, etc). I have developed a very quick shorthand for this and shortcuts for devising new techniques on the fly. So then maybe the player goes and fights this guy. How that goes is going to be down to tactics, styles and dice....and the outcome could really make a big difference: the player winning might mean they grow their reputation, if they really trounce the person he may even want to become their pupil. Or maybe the player gets destroyed and the reverse happens, or this is the start of a lifelong grudge (possibly a friendship). It is all going to depend on the back and forth details between the player and the NPC.