wetware: Your expected numbers of the different types of responder roll look right to me.  Using them, I get a chi-squared statistic of 274, when I'd  expect a figure of 13.8 or above for only 1 in a 1000 samples (years in this case) if the dice rolls were totally independent and generated fairly.  So we're talking odds of many, many millions to one against this being the case.

I think by now that most of us are agreed on this being caused by non-independence of the opening rolls rather than non-fair "dice" being used.  But data and stats are fascinating, so feel free to produce more!

Hopefully some of your analysis will give someone some insight as to when at least part of the opener's roll is used as part of the responder's roll too.  My guess would be that the actual rolling is simulated and so re-rolls will be generated if both players are assigned the same first roll, and it's this re-rolling that isn't working properly.  I think pB's already suggested this.  Hard to test though ....

Resher: Why would anyone want to re-roll if the first rolls were the same? And what do you mean by "simulated"? I personally never had a vision of Fencer rolling an actual pair of dice every time I click ...