The answer, before the Renaissance, is not really. Firstly, broadly speaking, prior to the Renaissance it was the imperial image of Rome that prevailed in European political thought. To give two examples: after the revival of the Roman legal tradition, later medieval legal scholars tended to focus almost exclusively on Justinian's Code, with a dominant school called the Glossators taking this Code to be the definitive basis of all civil law. We can also see in places like Dante's tract On Monarchy defences of the idea of universal monarchy with imperial Rome firmly in view -- Dante argues, for instance, that God's decision to be born as a man under Roman imperial rule proves the empire's greatness. It's only with the Renaissance that Republican Rome displaces the empire as the main object of political thought.

The Roman legacy before the Renaissance was divided between Pope and Emperor, an intellectual and political struggle that left little room for articulate republicanism, albeit republican sentiment certainly existed in Italy. As historians like Quentin Skinner (The Foundations of Modern Political Thought) have suggested, it was precisely the opening up of this little room from about the 13th century by people like Bartolus of Saxoferrato which laid the foundations for modern political thought.

From the middle of the 13th century on, moreover -- when these texts started becoming widely available again in Europe -- medieval thinkers tended to rely on a Greek, rather than a Roman, source for their understanding of republics: namely, Aristotle's Politics. In this context, although references to the Republic were often made, it's difficult to see it as having played any really substantial role in influencing how people thought about politics. As Fergus Millar notes:

What does not seem, so far as I can discover, to be present [in late medieval political thought] is either any detailed discussion of Roman institutions or any systematic analysis of the Republic as a political system using the categories provided by Aristotle ... still less ideas derived from these as to the desirability of a mixed constitution. (Millar, The Roman Republic in Political Thought, p. 58)

In the specific case of Venice, it's important that in contrast to places like Florence there was actually not really any attempt to develop a systematic defence of its specific institutions at all until rather late. The pre-eminent spokespeople for Italian republicanism before the Renaissance and for much of that period itself were from other places -- people like Ptolemy of Lucca, Bartolus of Saxoferrato, Marsilius of Padua, and of course eventually Florentines like Guicciardini and Machiavelli.

This is perhaps because Venetian political institutions developed, virtually uniquely among the major Italian cities, on an evolutionary, customary basis rather than through instantaneous upheavals and specific design. One apposite example of the difference here can be seen in the fact that while many Italian cities instituted 'consuls', an obvious if superficial Republican reference, Venice by contrast always had its 'doge' -- a form of the Byzantine title for an imperial governor, dux.

Even once the Renaissance intellectual revolution was underway, Venice was often compared against rather than alongside the Roman Republic. The problem with imitating the Republic is of course that, in the end, the Republic obviously failed as an institution -- a fact brought up by Paolo Paruta, maybe the pre-eminent Venetian historian and political thinker of the Renaissance and a man who followed firmly in Machiavelli's footsteps. Paruta argued that Venice was superior to the Republic because it was less democratic and therefore more stable.

Finally, even after comparisons between Venice and Rome became more widespread, and Venetians did perhaps see themselves more in the footsteps of Rome, I'm nonetheless not sure that Venice ever really attempted to 'replicate' Roman institutions.