What is a quantum coin toss?

Is there an error in this video 5:35? She says that, if person A had been cheating and had in fact used diagonal filters, person A could prove her innocence by reading off the "diagonal" column. But the diagonal column of the person B will not match that of person B, because person B will may wrongly record vertical photons (coded by person A as 1s) to be 0s if they don't pass through -- assuming we're using the 1-valued diagonal filter -- or to be 0s if they do pass through -- assuming we're using the 0-valued diagonal filter. So there's no way person A could confidently match person B's data based on her own. 


Presumably, then, person A can only read off the rectilinear column as a kind of proof. Which works if person B guesses diagonal and wants to know they weren't duped; person A would clearly only know the rectilinear data if they had knew the state of all the photons themselves. But consider the opposite case: person B guesses rectilinear when person A used diagonal and wants to know she hasn't been duped. Person A *can't* -- with certainty -- read off person B's diagonal filter because of the probablistic nature of diagonal polarization. (And person A clearly can't simply proclaim that they don't know the rectilinear sequence; that implies person A is trustworthy, which is the very assumption we're designed this experiment to eliminate.) So Person B can't be reassured by a correspondence of binary subsequences, and we again destabilize into distrust.