Depends what you mean by "the people." There was public opinion and it had a very strong influence on the realities of power all the way until somewhere in the middle of the Time of Troubles (3rd Century CE). This was because the Roman upper classes had considerable influence even in the Empire -- Emperors who governed in defiance of the Senate often found themselves falling to Senatorial conspiracies. Witness the fates of Caligula and Nero, in particular.

Now the Senatorial class had connections by blood, friendship and patronage to the middle classes, the Equites or "Knights." These in turn had connections, and these connections all the way down the social ladder -- and a lot of what went on in Rome was a matter of politics communicated through these webs of connections. Wives and to some extent even some slaves had influence in this network.

Thus, even though neither the Roman Republic nor the Roman Empire was "democratic" by modern standards, they were most certainly res publicas by the standards of the Ancient World. The real changes come with (1) the transition from Republic to Empire, which renders both Senate and Assembly less important, and (2) the 3rd-century Time of Troubles, which makes the key to power purely the control of a big loyal army.

From the Time of Troubles on, "public opinion" in the broader sense became less important, because increasingly a disarmed populace was helpless before the demands of this or that jumped-up general with an army willing to kill any number of dissenting civilians to please their commander. Even the great Emperors of the 4th century onward, such as Constantine the Great, were just exceptionally _competent_ commanders of said loyal armies -- they were no longer really "public officials" in the sense of the Early to Middle Western Emperors.

- Jordan