There were not "branches" of government in the modern sense with the judiciary separated from the legislature and the executive. But Ammianus is not correct that the Senate had "all" the power either. The Senate appointed the Consuls each year (the executives, who effectively took the place of kings) and pretty much all the other officials. Senate resolutions (consults) had virtually the force of law. The Senate itself could act as a judiciary over its own members, particularly in cases of treason and such, and the officials they appointed had judicial as well as executive powers in their respective jurisdictions.
But there were also other, broader assemblies, of the army, the citizens, and the plebs (the membership of which would have overlapped a lot), and each of those had genuine powers. Formal written laws (leges) had to be voted in by the citizen assembly - they couldn't be simply decreed by the senate, and they had more weight than senate consults. Perhaps most importantly, the tribunes of the plebs had veto power over the acts of any official, which was a protection of the rights of common citizens against abuse by patricians.
The whole thing had begun with revolt against the abuses of corrupt kings. The senate had probably been a council of nobles advising the kings before that. With the kings gone, the senate took control, and the appointment of consuls was a way to have someone fill the roles kings had played like leading the army. Over the first couple of centuries of the republic, there was great civil strife between patricians (nobles in the senate) and plebeians (commoners), particularly over burdens of military service and taxation. The outcome of it was the growth of the plebeian assembly's powers and the tribunes, and the creation of a system that would admit leading plebeians as well as patricians to the senate's ranks through public service. So it all evolved in response to the demands of the time.(Answer not mine)
