The senate was not a representative body, nor was it a governing body. The senate was an advisory body made up of former (and future) magistrates, whose job it was to issue consulta--which were not laws, although sometimes senatus consulta were carried by magistrates to the citizen assemblies as bills and passed into law--to the various elected magistrates that ran the city.

Such records did exist, sort of. Various magistrates and collegiate bodies posted their acta in the form of alba, tablets posted in the forum or nearby their meeting places. The oldest of these were the various priestly alba, but the praetorian album posted the urban praetor's notices, which had some legal importance--we're talking essentially of the results of legal cases. During his consulship in 59 Caesar forced the passage of a law publishing the acta senatus and the acta populi, the doings of the senate and people, respectively. Posted daily these were, especially the acta populi (following Augustus it was forbidden to publish the acta senatus), essentially catalogs of the various alba. We have no trace of any surviving alba or acta, and it's questionable to what degree they were ever meant to be preserved. We have some references to them, and a satiric version is presented in the Satyricon:

VII kalendas Sextiles: in praedio Cumano, quod est Trimalchionis, nati sunt pueri XXX, puellae XL; sublata in horreum ex area tritici milia modium quingenta; boves domiti quingenti. Eodem die: Mithridates servus in crucem actus est, quia Gai nostri genio male dixerat. Eodem die: in arcam relatum est, quod collocari non potuit, sestertium centies. Eodem die: incendium factum est in hortis Pompeianis, ortum ex aedibus Nastae vilici.

July 26: At a Cumaean estate, belonging to Trimalchio, 30 slave boys were born and forty girls. Five hundred thousand modii [the number is comical--milia quingenta can mean literally five hundred thousand, or it can be "a buttload," some enormous uncountable number] of wheat was taken from the threshing-floor to the granary. Five hundred oxen were broken. On the same day: Mithridates the slave was crucified, because he cursed the genius of our master Gaius. On the same day: ten million sesterces, which could not be invested, were placed into the safe. On the same day: there was a fire in the gardens at Pompeii, having arisen in the house of Nasta the overseer.

Petronius from there includes the aediles' edicts, a list of wills that Trimalchio was excluded from, punishments inflicted on Trimalchio's slaves, etc. This was the stuff of the acta populi. The imperial household also produced its own extensive records--Suetonius had access to them. But, like the private records of countless households throughout the Roman history, these are generally not preserved. Why should they, after all? For what purpose would precious resources and manpower be wasted copying them? Sometimes the contents of these imperial records are recorded in, for example, the jurists, and especially in late antiquity we know of more than a few products of the imperial household, which came to be written in distinctive scripts only known to the imperial administration. Similar records were kept at least at some military posts, the only surviving example of which are the Vindolanda Tablets. Administrative records, public and private, are among the more common papyrus fragments at sites in Roman Egypt, but these are fragmentary and most have never been read, let alone published.

There were two bulletins of public records that were published in the Late Republic: the acta diurna and acta Senatus. We don't know a ton about these because they don't survive as documents themselves, and when they get mentioned they get mentioned in ways that make it hard to tell what's in them.

The acta diurna was the older publication. It seems to be a record of public events, including public speeches and court cases. The major source for these (although they are mentioned in several places - for instance Cicero seems to have sent copies to his friends away from Rome to keep them up on the news) is Asconius. He lived in the first century CE, and wanted to write a commentary for his son on Cicero's speeches, and referred to the acta for legal notices such as arraignments, indictments, etc, legal teams, and verdicts, for court proceedings. He also attributes to the acta notices for decrees of the Senate, contiones (public meetings), and details like what day Milo left Rome for Lavinium (important to establish for a murder trial). That Asconius used them means someone kept an archive of old editions. What's bizarre is, as far as I know, Asconius is the only Roman who used the acta to inform his reading of Cicero's speeches, despite interest in Cicero being perennial.

The acta Senatus Caesar established during his consulship in 59 BCE, and were the proceedings of the Senate. There were public under Caesar, but Augustus forbid their publication; they were important - Augustus appointed a senator to be responsible for them, but they were not to circulate beyond the Senate.

There's a bit of confusion between these two publications. For instance, the OCD makes these two different publications, the acta diurna being older, and the* acta Senatus* being instituted in 59 by Caesar, while Brill's New Pauly mentions the acta diurna (also called other things), but not an acta Senatus, and claims the acta diurna was established probably in the time of Caesar ("Probably since the time of Caesar (Suet. Iul. 20,1) they serve as the regular publication ( album ) of official news..."). So we don't seem to have a very good grasp of what was in these documents or how they were related.