I went by myself in 2015 after a night at the Conrad. Ended up wandering through the inner market, but didn't go to the auction. Had a lovely exchange with a man at one stall who asked me if I was from America, I said Australia, and he pointed at the large tuna he was cutting, and said it was from Australia (he was speaking in Japanese, I was speaking in English).
Went back with Mr Katie and Little Miss last September. We didn't go to the inner market, but ended up in the proper outer market area to buy a knife, stone, and then have sushi lunch. Mr Katie was happy with his knife, kept trying to buy another, but the lady at the knife stall kept insisting that he didn't need any more!! What a strange retail experience.
We then had sashimi lunch, which Little Miss wasn't keen on, and spent most of the time playing with a baby at the table behind us. She gave the baby one of those little koala toys - the grandparents came back a few minutes after leaving the restaurant to give Little Miss two packets of gorgeously decorated bandaids. (I realised on that trip my great idea of letting Little Miss give the kids she'd play with the little koalas turned into a reciprocal giving situation, which I hadn't anticipated).