I could imagine a number of places where partisan coordination might have occurred, including in the selection of names to be drawn from or in the negotiations over legislation/policy selection.
And what should I read to learn more about the politics of Renaissance Florence? I'm particularly interested in the period before the rise of the Medici, say 1328 to 1434.
I think that rather than political parties, factions might be a better term to use. The Guelphs and Ghibellines are probably the closest thing to modern political parties. They emerged in a number of northern Italian communes in the 12th century. They are the Italianized terms for partisans of the Welf Family of Bavaria and the partisans of the Holy Roman Emperor who held the fortress of Waiblingen (Italianized as Ghibelline).
These factions within communes were either for the Emperor and his designs in Italy (Ghibellines) or for the Pope and resisting the Emperor's incursions (Guelphs).
By the fourteenth century (and a good introduction should explain this), the Guelphs had separated into the Black and White Guelphs who were for or against the Papacy. Dante was exiled as a result of the political turmoil in the city during his life.
We should also consider familial factions within the city. The Medici were newcomers to political life in Florence. The Pazzi who attempted to murder Lorenzo and Giuliano d'Medici were an old family. They had a representative at the fall of Jerusalem during the first Crusade (something that was respected). This family gave up titles in the fourteenth century to hold political office. Although they would have joined up with a faction of the Guelphs at this point, they would have also been part of a familial network.
In addition to Guelphs and aristocratic groups, the guilds of Florence were politically active. 6 of the 9 priors of the priori of the signoria were selected from the leading guilds. These organizations were incredibly wealthy and commissioned many great public works of art in the fourteenth and fifteenth century. Indeed, the calimala (cloth finishers guild) paid for Ghiberti's beautiful contribution to the Baptistry of St. John.
Finally, in 1378 there was a revolt of the ciompi or wool carders guild. This guild represented the lowest echelons of society included non guild laborers. They briefly installed a popular government.
This is just a cursory discussion of the members of the Florentine government. Najemy's history of Florence is the best history of the city during the Middle Ages available in English.