So we took our passport, Permesso di Soggiorno, Certificato di Residenza and three passport-size photos to the Ufficio Comune (on the ground floor of the Palazzo Vecchio). We also took a friend along because she claimed that we needed to have a witness, but it turned out that as long as our paperwork was in order a witness wasn't necessary.

At the office, we took a number at the front desk and waited to be called to a "sportello". We waited about 10 minutes for our turn, then we waited about another 15 minutes for all the paperwork to be drawn up - and we walked out with two brand new Carte d'Identita! It was so easy and now we've bought a car!