Arnold's account of this debate it (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 in 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 2013 in Institute Poincare.

Then he describes Serre's speach and his own.
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.

My impression is that I read somewhere also Serre's talk. But I cannot find any trace of
it in the Internet.