The Roman state had no such thing as a "legislative branch." The senate had no powers of legislation and was legally a strictly advisory body composed of former magistrates (by Sulla's time above the rank of quaestor). Although the senate reserved the quasi-extralegal power of the senatus consultum and especially the so-called senatus consultum ultimum, both advisory notices to individual magistrates with varying degrees of compulsion and seriousness, the senate held no right to pass legislation. Both elections and votes on legislation were passed by the citizen assemblies, particularly the centuriate and tribal assemblies, after being brought before the assembly by an elected magistrate.

Theoretically that would mean that "the people" would have had both great participation in legislation and great knowledge of it. In fact, however, that seems not necessarily to be the case. The property classes of the centuriate assembly, which gave vastly inordinate voting power to a small group of very wealthy individuals, is more or less irrelevant to the legislative question, since the centuriate assembly did not normally vote on laws. However, it gives an indication of what the Roman conception of citizen assemblies was like--it was not necessarily formally equal. The tribal assembly, which did most of the legislative voting, either strictly as the tribal assembly as a whole or as the plebeian assembly, which was based on the full tribal assembly, contained only four urban tribes out of its total thirty-five. Moreover, voting always took place solely in the city, and while rural inhabitants could travel to the city in time for the (fairly) regular electoral votes (though sometimes the agricultural season prevented even that, as during the vote on Ti. Gracchus' second tribunate), legislative votes could occur more or less at any time, and probably members of the rural tribes could not be at hand in great numbers during ordinary legislative procedures. The "average" urban dweller likely had little ability or inclination to participate in electoral politics. Most urban dwellers were semi-itinerant day-laborers, fighting a constant uphill battle first of all to find work for the day (far from guaranteed) and then to keep themselves fed and sheltered for the day in the expensive conditions of the city. Far from the old view of the "idle plebs" the Roman plebs were a working population, one that was laboring pretty much constantly. Urban day-laborers had little opportunity to read (if they could read) the various urban acta posted in and around the forum that summarized political and other domestic activities (and, after Caesar's consulship in 59, the meetings of the senate) as they bustled around the city looking for work, and they likely learned about political activity by word of mouth. That's not to say that rumor wasn't a highly important piece of politics--as Ray Laurence (despite his absurd claim that all urban workers belonged to a collegium) notes in "Rumour and Communication in Roman Politics" in a pre-industrial society rumor (which is to say reports of events or speeches second-hand) is probably the most important means of communicating information--but likely information that either didn't concern them or wasn't especially pressing was often ignored by the urban plebs. And in any case, few urban workers, not unlike their rural counterparts, had the ability to attend even the contiones of speakers, let alone the legislative assemblies. The need to find work for the day or starve was paramount, and a day of work lost to go vote was a day one might not eat. To this Mouritsen has added spacial and temporal considerations in the ability to get large bodies of people into the vote, and we're left with a picture of a Republic in which really only a very small number of people were actually voting under normal circumstances. Which is not necessarily to say that the Roman people were disinterested. The possibility of urban workers without work for a day acting in a political capacity after they've found nothing better to do is noticed by Mouritsen, but largely ignored--it's been picked up by other scholars, in particular as one of the probable ways that violent politicians recruited temporary muscle. So we should not necessarily have an image of a totally politically disinterested plebs during the Republic. Rather, the political system and the realities of economics and setting provided little incentive at most time for the common man to show up at the vote.

However, since you mention the "Roman Empire" I must assume you're talking about the Principate. Since 59 BC the acta senatus had been published openly, which theoretically meant that everyone (or at least everyone who could read or who heard their contents reported by someone else) knew what was going on in the senate daily. This right, however, disappeared under Augustus, who ordered that the acta still be recorded but no longer published them publicly. Still, there were means in the Principate as in the Republic for the urban plebs to find out what was going on. Clients of politicians frequently picked up and spread news of political activities, especially since clients were an important (if often overstated) means of political communication. Moreover, Roman politicians did not generally ignore the populus entirely, although what the populus consisted of at any given moment is a question that's been discussed rather a lot recently. Politicians gave contiones, informal speakers' assemblies, not unlike political rallies. The exact makeup of these contiones is pretty uncertain (they seem more often than not to have been made up of a politician's own supporters, and often probably included rather a lot of the senatorial class and equites), but the open promulgation of information outside the space of the curia, which was even ideologically linked to communication with the populus, is hardly unimportant. Contiones were less common in the Principate, as the emperors severely limited the legal and expected independence of members of the senatorial class, but speeches continued. Speeches in the senate regarding important urban matters were very frequently followed by speeches ad populum on the rostra. Cicero, for example, delivered the Third Catilinarian ad populum after a senatorial session, which it mostly summarizes. It should be noted that formally speaking the citizen assemblies were still in charge of legislation in the Principate. Under Augustus in particular the elections and legislative votes continued, although under very controlled circumstances. Tiberius, however, did away with voting for magistrates, and from his time on the assemblies did not pass laws, although theoretically they still could. The Republican system of legislation through the tribal assembly or by plebiscite was still preserved to a limited degree under Augustus, but already by his time the quasi-legal status of the senatus consultum was already being enforced. In the Republic senatus consulta were not legally binding, although they often had the force of laws, especially when they were being supported. For example, in 61 BC Metellus Celer, though only a consul-designate, used his towering auctoritas to bully an unnamed tribune who had attempted to celebrate the ludi Compitalicii against a previous senatus consultum. The tribune in question had every legal right to celebrate the ludi Compitalicii, since the senatus consultum was not legally binding, but nevertheless he was prevented from doing so anyway (though Metellus had no actual powers to stop him!). Since leges were not being promulgated anymore much "legislation," if we want to be a bit free with terminology, went through the senate through the right of senatus consultum. More importantly, however, the edicta of the magistrates, which were originally not unlike the senatus consultum in that they were formal pieces of advice issued by the elected magistrates but had no legally binding status, became in the Principate an important means of passing legislation without the need for the citizen assemblies. The emperors issued edicts, which while not technically leges generally were in no significant way different in practice. Indeed, the status of edicta became so clearly understood as leges in practice that the terminology of the lex was taken over by edicts. In the famous edictum Diocletiani the word edictum is never used, although the verb dico ("I say," used usually in edicts, since they were pronouncements by the magistrates) does, and the text actually calls itself a lex, though it was legally speaking no such thing! The jurists treat senatus consulta during the Principate and the edicts of the emperors as more or less equivalent to laws, although since emperors could issue edicts more or less at will and for specific events the jurists sometimes get into trouble when emperors issue conflicting edicts or issue edicts that are so specific as to cause problems with legal precedent.