I'm not sure I get the setup of this puzzle. He tells you at the start that the total between between you is 18 or 20. So you know from the start that your friend sees 6 or 8 and she knows that you can see 10 or 12. So neither of you 'learn' that the other person sees at least 1 / fewer than 19 on the first round - you both already knew this.
So what knowledge have you actually learned after the first round, that you can take forward into the second round?
So what knowledge have you actually learned after the first round, that you can take forward into the second round?