Arnold's account of this debate (in Russian) is here: http://www.mccme.ru/edu/index.php?ikey=viarn_burbaki&post=25742517_1256 and here http://vivovoco.rsl.ru/VV/PAPERS/NATURE/BURBAKI.HTM
Let me translate few words from the beginning.
Serre challenged me with this motivation: "I wanted to tell about the influence of Bourbaki on mathematics. If everyone will speak the same, especially panegyrics, this will be boring. Thus I started to search who would express an opposite opinion to mine. After having leafed through the list of mathematicians of the world, I understood that this must be you."
The duel happened on 13 of March 2001 in Institute Poincare.
Then he describes Serre's speach and his own. According to Arnold's account, the published version of Arnold's talk during the debate is called "Mathematics and Physics: a parent and a child, or sisters?" English translation: Physics-Uspekhi, 1999, 42, 12 1205-1217. Here is a free version: http://ufn.ru/en/articles/1999/12/c/
However, comparison of dates shows that this cannot be the true text of Arnold's talk at the Inst. Poincare. I reread the account of the duel (cited in the beginning) and on my opinion 90% of what Arnold says there (about Serre and about other things) is just not true. Unlike this account, the paper in Physics-Uspekhi looks reasonable.
My impression is that I also read Serre's talk somewhere. But I cannot find any trace of it in the Internet. So all we have this biased (to say the least) account by Arnold.