The debate was oral. But here is Arnold's written account of it (in Russian):
http://www.mccme.ru/edu/index.php?ikey=viarn_burbaki&post=25742517_1256

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 decribes Serre's speach and his own.
It is my impression that I have seen both talks in written form. But I cannot recall where.