Unfortunately I cannot answer your question directly.

In the West, very very few people read Greek and there was hardly any Greek for them to read, so any notion of Greek democracy they had came from Romans who generally thought Athenian democracy was wacky (as did most Greeks). They are used by Cicero and others as an example of the disadvantage of popular participation. The Romans viewed it as a danger leading to mob rule and violence.

What I can address is the way Roman ideas of government influenced the Holy Roman Empire.

We don't usually think of the HRE as a democracy or a republic, but the Germans did. They elected their emperor for life at a diet of all the lords of Germany. They viewed this election as a popular election that gave the emperor a mandate from the people. This is in direct imitation of the lex de imperio, an acclimation from the people of Rome to give the emperor specific powers. We have a surviving example called the Lex de Imperio Vespasiani.

So wrapped up in this vote is a notion of popular sovereignty. The people then vote to give the emperor their powers. The Germans imitated this, except that their notion of "people" aka citizens was limited to the nobility. This can be seen in some of their language. The word usually translated as "diet" is curia, which has its root as a deliberative body of citizens. They also used the word comitia, a elective assembly in ancient Rome to describe deliberative groups in medieval Germany.

In the 12th century, Frederick Barbarossa marched on Rome because the Romans would not acknowledge his authority over them because they had not given him their powers by a formal lex. He responded by saying that the Roman people (German lords) had given him their authority by vote, and that the City was thus his possessio, a term used specifically for property under his full authority. This is detailed in Otto of Friesing's Deeds of Frederic 2.30. This argument rests not only of Roman traditions, but also on Biblical notions of empire, and Medieval notions of nobility.