Ah, I should not have stuck with the original post's use of the word "pass". I was envisioning each word being announced publicly, and the speaker designating the next person with eye contact or pointing or something.
In this way, everyone does have their own stack and if their hearing and memory are perfect, everyone has the same stack contents. In practice, memory corruption causes the players' stacks to disagree more at the bottom as time goes on, but agree reasonably well toward the top. If a word is misheard, some players might even disagree about the current value at the top of the stack.
Since they used the word "pass" maybe the OP does mean it's private, in which case it's a pretty different game than what I'm guessing at here. In my version, during the second half of the game, the current speaker could designate any player to be the next to pop and announce their top of the stack.
If the passing of words is private, then players need to remember who they interacted with at various points in the first half so they can try to form those same pairings again in reverse. And I guess the signal for the mid-game reversal point would require whispering the word they just got, back into the ear of the same player who just gave it to them.
Edit: Ohhh, I neglected the fact that they "circle up" in the OP. That solves the problem of getting the pairings right. You know exactly who you interacted with if you're all in the same spots in the circle and only interact with your neighbors. Okay, the word passing must be private. Which makes sense if you're hanging out behind the curtain at a theater and you need to be quiet. I wonder if the game that I've imagined would be any good.