Well, that's kinda a loaded question. Here is what Cicero said about good government in the time shortly after the establishment of the Republic (De Rep. 2.56):

though the people were free, not much was done through them; most things were done on the authority of the Senate according to custom and precedent. The consuls had power which, though limited i length to one year, was in its nature and legitimacy equivalent to that of a king. And (a factor that was perhaps the most vital in maintaining the power of the aristocracy) the rule was staunchly retained whereby the people's corporate decisions were not valid unless endorsed by the Senate's authority.... Yet, with the people's consent, all the power was firmly held in the hands of the leading citizens...

This comes from a fictional dialog set about the year 129 BCE, in which the character Scipo Aemilianus (who was a real person) describes the history of Rome among other things. Cicero explicitly calls this a "mixed constitution" (a term which is difficult here because other people used it to mean different things). For Cicero, this form of government worked because people were allowed to express their gratitude to aristocrats for their, well, awesomeness, by electing them to positions in which they exercised power, with the advice of another body of aristocrats in the form of the Senate. The people pick the person to do the doing, the Senate helps decide what ought to be done, and the person does it. Perfect government, right? Where would that fall on your "democracy for the people/unelected aristocrat" dilemma?

In the 20th century, largely influenced in the Anglophone world by the work of the historian Ronald Syme, the democratic elements of this setup have widely been considered to be a sham. The people voted, for candidates selected by the aristocracy, in assemblies that placed far more weight on the opinions of aristocrats than on the poor, and the poor were limited in addition by social obligations imposed on them by ties of clientele, especially in the time before the introduction of the secret ballot. Syme either extrapolated this view of Roman history to all human history, or imposed a view of Roman history he developed elsewhere, and formulated along these lines:

In all ages, whatever the form and name of government, be it monarchy, republic, or democracy, an oligarchy lurks behind the façade....

A careful reader will pick up weighted terms here too; in Aristotelian political thought oligarchy is the devolved form of aristocracy. I have never thought about this before, but maybe Syme meant us to think of the Aristotelian Pure Constitutions here.

Anyway, under this model, there WERE elections, but of candidates pre-selected, and conducted in ways that made sure (mostly) that only the right people got in. There is some evidence to support this; over the course of the Republic (some 500 years) only a handful of outsiders (Novi Homines or New Men) ever reached the consulship. TP Wiseman has a book, New Men in the Roman Senate, that discusses these, and if I remember correctly he could find only 15 men out of something like a possible 1000 consuls that made it from outside the aristocratic circle.

More recently, following PA Brunt, people have been thinking the democratic elements of the constitution were stronger than Syme and his buddies believed. However votes were weighed, the fact is that politicians still tried to convince people to vote for them. The Commentariolum Petitionis is what pretends to be (and might very well be) a manual by Cicero's brother Quintus on how to run a political campaign, and Quintus is very clear you have to get the support even of the poor. Politicians were brought up on charges for illegal electioneering. From time to time foreigners were expelled from the city during elections. A secret ballot was instituted (Cicero hated this - it made it impossible to see if your bribe generous consideration for a valued friend worked won deserved appreciation) It's hard to imagine why these things happened if the voting didn't matter.

Here is the select bibliography from a paper on this more recent position I gave a month or so ago:

Alexander, Michael. Trials of the Roman Republic: 149 BC to 50 BC, Toronto: University of Toronto Press, 1990. Lintott, A. “Electoral Bribery in the Roman Republic,” The Journal of Roman Studies, 80 (1990), pp. 1-16. Morstein-Marx, R. “Publicity, Popularity and Patronage in the 'Commentariolum Petitionis',” Classical Antiquity, 17 no. 2 (1998), pp. 259-288. Wallinga, T. “Ambitus in the Roman Republic,” Revue Internationale des Droits de l'Antiquité 41 (1994), pp. 411-442. Yakobson, Alexander. Elections and Electioneering in Rome: A Study in the Political System of the Late Republic. Historia Einzelschriften 128. Stuttgart: Franz Steiner Verlag, 1999.

So, how democratic or aristocratic Roman government was is currently up for grabs. There are still a lot of people (I learned when presenting this - I thought they were all dead now) that buy the aristocratic argument; the more democratic argument I think is still being developed but seems to me to be ascendant.

Whether EITHER of these models represents government for the people, well, I suppose you must use your own political judgment. Orwell once noted that in modern English "democracy" has little meaning other than that we're meant to feel good about it, and even "democracies" meant strictly as a descriptor of a form of government are often criticized for colossally stupid decisions. I think just about anyone might find an example or two coming easily to mind.