But you already know your friend sees 6 or 8 based on what you can see, and that you know that between you, you see 20. Both of you knew already, that this was not a possibility. I think I'm missing something from the setup.

You know that your friend sees 6 or 8 and you know that if your friend sees 6 then they know you see 12 or 14 and (you know that they know that) if you see 14 then you know that they see 4 or 6 and (you know that they know that you know that) if they see 4 then they know that you see 14 or 16 and ((you know that they know that)^2) if you see 16 then you know that they see 2 or 4 and ((you know that they know that)^2 you know that) if they see 2 then they know that you see 16 or 18 and ((you know that they know that)^3) if you see 18 then you know that they see 0 or 2 and ((you know that they know that)^3 you know that) if they see 0 they know you see 18 or 20 ((you know that they know that)^4) if you see 20 then you know the correct answer and it's over

So when it's not over on the first day, that updates your knowledge about each other's knowledge (about each other's knowledge ...) by eliminating that particular possibility. That goes on until only one possibility remains and you know the answer.

Ah, nice. This is the answer I was looking for.

This bugged me too, the process works but they’re throwing away a lot of information at the start, which should mean it can be done faster.

I made another answer that works the same way but doesn’t throw that away, and it seems it’s possible to finish on Day 2.