For the first question: if I observe multiple coin-tosses and count what fraction of them are tails, then what should I expect that fraction to be? (Answer one half). Clearly "I" here is anyone other than Beauty herself, who never observes the coin-toss.

For the second question: if I interview Beauty on multiple days (as the story is repeated) and then ask her courtiers (who did see the toss) whether it was heads or tails, then what fraction of the time will they tell me tails? (Answer two thirds.)

What information is needed for this? None except what is defined in the original problem, though with the stipulation that the story is repeated often enough to get convergence.

Incidentally, these questions and answers aren't framed as bets, though I could use them to decide whether to make side-bets.