How would it change? If I write my thoughts on the size of a group (which we both may well disagree on) or whether some tricky situation might in fact be dead I am just cluing my opponent on my thinking, which entirely defeats the purpose of any and all note-taking. I don't see how using a paper extension of your own memory necessarily changes the nature of the game. It's still your own private thoughts, just with a little reminder.

I think taking notes is very different from the nature of the game. When two people play go the only equipment are the board and stones. Over time as people wanted to improve, they would record games to ask others where they could get better. The intent is not to use paper and pencil for anything but game recording.

Do you think people should be allowed to play out the game on a second board, as well?

I would be wary of drawing parallels here. Go is a perfect information game. Magic, by design, is not.

Yes, but only if the board is a tattoo and the stones are pocket lint.

Next time I will have some colored rings on my fingers. I'll turn them to represent the score of the groups. Right hand my own groups, one finger each. Left hand the opp's.

I agree. You develop the ability to memorise notes better if you keep track in your head, and good habits become second nature over time.