How collaborative should Hx be?

I don't have a lot of experience with Apocalypse World. I wonder how much Hx is meant to be a collaborative process. Do both of the affected players deciding things about the choices that set Hx with each other? Or if it's one player's turn, do they just get to define everything and make that stand?

It depends on your group, generally. The rules give the authority to the person choosing, but that authority only extends to what the Hx question says -- the details and context are usually the result of collaboration between the other players involved (including the MC.) So the Hardholder can say that PC X stole something from them or betrayed them, but they don't necessarily get to decide which, or what the circumstances were.

My personal take is that the Hx setup gives the player choosing the authority to initiate something, unilaterally, but that ultimately the group and the other player need to confirm and engage with it before it really counts. Both sides of this equation are important -- the connection needs to initiate with the first player because the point is to give them a connection that is interesting to them, and it needs to be confirmed by the other affected players because otherwise it's not going to be much of a connection; they're going to be the only one of the two who thinks it's a cool thing to play around.

I stress the importance of both sides just because one extreme point on the 'how groups can approach Hx' is for the player whose turn it is to be like 'well it says here somebody betrayed me -- do any of you want to be that person?' Which is fine, if they honestly don't have a preference, but not good if they do actually kind of hope it's gonna be PC Y, but don't feel comfortable just saying so for whatever reason.

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Also, apparently in the new edition, this aspect of Hx has been totally overhauled, but I am not aware of the particular details.

It also varies by playbook. Most of the Hx options are stuff that reflect your opinions of other characters ("you think X is trouble," or "Y fascinates you"), and these don't really require any input from the other player.

Some are more involved ("X left you in a bad spot once"), and in those cases it's good to work together to figure out how and why. We had one of these in a recent campaign, where it was determined that the Battlebabe had left the Operator hanging, but that the reason she had done so was because the Faceless (whom we had established was part of the Operator's crew) had gone off the deep end and the Battlebabe didn't want to be part of the inevitable massacre to follow. So two players (the Operator and the Battlebabe) got to exercise Hx options from the same fictitious "past event," and it resulted that much of the distrust between them was over a misunderstanding. That informed their interactions for quite some time.

But some playbooks allow you to just straight-up say things about other peoples' characters. The Skinner in particular is awesome for this. "Player X is your lover." Boom. You don't need their permission to say this. So your super macho, manly-man Chopper who beats his biker cronies into shape? Yeah, he's banging my willowy, snappy-dressing dude with a lovely singing voice. And it's up to everyone involve to figure out how this plays out in the game. Does it add a complication (the Chopper's gangers give him crap for being gay)? Or is it not even a thing? Or is it a different kind of thing (Ace, the Chopper's most capable lieutenant, covets the Skinner and wants him for himself)?

One of the beauties of AW is that you are not creating (or even playing) your character in a vacuum. Due to the Hx mechanics, even something as fundamental as your character concept is open to input and influence by the other people at the table, which is almost always hilarious.

My personal view of Hx is that its from one player's perspective, and coloured by that. For an example, the Gunlugger has "One of them once left you bleeding and did nothing for you" as a Hx option and that's what the Gunlugger saw happen. She doesn't know/care that you were being chased by Jackals boys and couldn't stop, all she cares about is that you left her!

However in the new edition Hx is indeed reworked, I wont give too many details because if you're interested you can send a little bit of money Vince's way, but it does give each person the power to decide how they're known by Hx.