We cannot be certain whether the Romans borrowed from the Greeks, or if they did, how much they borrowed. In my opinion, the Romans did borrow and learned from the Greeks more than the Romans would like to admit. Three points leads me to this conclusion.
First, Greek cities flowered before the founding of the the Republic. For example, Athens in the 400s reached its peak. The Roman Republic did not start until 509. Moreover, the Greeks were in southern Italy, and so the Romans could have easily learned much from them.
Second, it is interesting that the Romans abhorred kings. This was something that they shared in common with Greek democratic cities. Apart from a mythical king who was half serpent and half man (Cecrops) the Athenians did not have kings. They had a radical democracy, especially under Cleisthenes, where power was given to the people and diffused. The Roman constitution, likewise, is based the understanding of a diffusion of power.
Finally, we can say that the Roman constitution is a mixed form of government. They had a senate, magistrates, and two assemblies that voted. Loosely put, they had a democracy, oligarchy, and semi-kingship (with the consuls). This is very similar to Sparta. Sparta is not a democracy, but there were democratic elements in Sparta.
In light of these points, it seems strongly that the Romans borrowed heavily from the Greeks. In the end, the Romans created their own government, but it was not created in a vacuum.