I think you're referring in the OP text to Peter Alfonsi. He actually wrote an internal debate between his past Jewish self, Moses, and his present Christian self, Peter (the narrative frames them as friends). There are also Jewish texts which are responses of Rabbis to Christian arguments, which seem to have been prompted by informal discussions between Jewish and Christian religious authorities in a rather cordial environment.

Anyway, this sort of thing is called a "disputation". A number of them occurred between Jews and Christians. Generally speaking, these were organized by Christians, often with the purpose of convincing Jewish communities of the veracity of their argument. Sometimes Jewish communities were actually forced to attend. The Jewish representative could not meaningfully attack Christianity, since governmental authorities were Christian, and attacking the church was a serious crime. Jews were forced to defend against Christian arguments, rather than making arguments of their own. So while it was a debate, it wasn't really a fair one, but it wasn't just for show either.

The results of these varied. The best-known one is the Disputation of Barcelona, where a Jewish convert to Christianity named Friar Paul debated the famed Rabbi Nachmanides (known in Jewish sources as the Ramban). There are two conflicting accounts of the events--both sides claim victory. But it does seem that Nachmanides at least was not defeated, and may've been successful, which shouldn't be terribly surprising since he was one of the most reknowned Rabbis of the period. While the Ramban was free to actually argue, as he made clear in his account, social and political norms still restricted the extent to which he could attack Christianity.

The disputation of Tortosa had a result more favorable to Christians. The Christian side was represented by another Jewish convert, though one who seemed to be more learned and better-prepared, perhaps learning from the mistakes of Friar Paul. What's more, the position of Jews was much more tenuous than before, and the implicit threat of violence may've contributed to both the Christian success, since Jews would've been reluctant to advance any sort of argument critiquing Christianity, and because it would've backed up their subsequent missionary efforts.

So yes, they did occur. They were set up by Christian authorities to attempt to convert Jews. While the Jewish side was unable to exchange ideas freely, they were actual debates.

sources:

• Chazan, Robert. Fashioning Jewish Identity in Medieval Western Christendom. Cambridge: Cambridge University Press, 2004.
• ---. Reassessing Jewish Life in Medieval Europe. New York: Cambridge University Press, 2010.
• ---. The Jews of Medieval Western Christendom, 1000-1500. Cambridge, UK; New York: Cambridge University Press, 2006.
• Frassetto, Michael, ed. Christian Attitudes toward the Jews in the Middle Ages: A Casebook. New York: Routledge, 2007. Print. Routledge Medieval Casebooks.