Here's how that goes. I flip 3 coins. Say I get 2 heads. My probability estimate for "there are 4+ heads total" is now 4/8 (the probability that 2 or 3 of your coins are heads). For the full set of outcomes I can have, the options are: (0H, 0/8) (1H, 1/8) (2H, 4/8) (3H, 7/8). You perform the same reasoning. Then we each share our probability estimates with the other. Say that on the first round, we each share estimates of 50%. Then we can each deduce that the other saw exactly two heads, and on the second round (and forever after) both our estimates become 100%. For all possible outcomes, my first round probability tells you exactly how many heads I flipped, and vice versa; as soon as we share probabilities once, we both know the answer and agree.

(Also, you're not using "confidence interval" in the correct manner. A confidence interval is defined over an expectation, not a posterior probability.)

I still don't see any version of this that's simpler than Finney's that actually makes use of multiple rounds, and when I fix the math on Finney's version it's decidedly not simple.