When a new law was passed, the senate often issued an acta urbis, essentially a proclamation to be distributed to population centres. In this way new laws might be communicated in summary or through extracts, though our admittedly very limited archaeological evidence suggests that full, unedited transcripts were commonly issued. These would then be communicated to the public in two ways. Firstly, there were people called praecones, heralds, who essentially acted as newsreaders. They would stand in public spaces and read out a summary of new laws. Secondly, the transcript was sometimes inscribed by those cities onto bronze tablets and publicly displayed so that people could read the "news" in full at their leisure.

The Tabula Clesiana is a well preserved example, granting Roman citizenship to the people of Trentino and surrounding tribal groups. The issue was that many people from the town claimed to be Romans and therefore governed by Roman law, but it was discovered that many people were actually of tribal ancestry, specifically the Tridentini. This wasn't a transcript of a senate meeting, but a legal ruling from Claudius. It's quite long, Claudius was infamous for rambling, but the this is the important bit:

I realize that this category of people has not based their Roman citizenship on a sufficiently well-founded source. However, since it is said that they have been owned for a long period of use [that is to say, under Roman rule], and that you are so fused with Tridentini by not being able to be separated without serious damage to the beautiful town... they continue to be in legal status who they thought they had, and the more so because several of their condition is said to pay service even in my judgment hall, and some even have been officers of the troops...

The most famous example of this form of tablet is the Lyon Tablet, a very long bronze inscription which permitted people in Gaul to seek Roman political office provided they met the requirements, also from the reign of Claudius. It records heckles (or possibly leading questions, as /u/LegalAction has pointed out) and mistakes made by the emperor, who had a stutter.

Claudius: "It is surely an innovation of the divine Augustus, my great-uncle, and of Tiberius Caesar, my uncle, to desire that particularly the flower of the colonies and of the municipal towns, that is to say, all those that contain men of breeding and wealth, should be admitted to this assembly."

[Interruption]: "How now? Is not an Italian senator to be preferred to a provincial senator!?"

Claudius: "I will soon explain this point to you, when I submit that part of my acts which I performed as censor, but I do not conceive it needful to repel even the provincials who can do honour to the Senate House. Here is this splendid and powerful colony of Vienna; is it so long since it sent to us senators? From that colony comes Lucius Vestinus, one of the glories of the equestrian order, my personal friend, whom I keep close to myself for the management of my private affairs. Let his sons be suffered---I pray you--- to become priests of the lowest rank, while waiting until, with the lapse of years, they can follow the advancement of their dignity. As for that robber, Valerius Asiaticus from Vienna, I will pass over his hateful name. For I detest that hero of the gymnasium, who brought the consulship into his family before even his colony had obtained the full rights of Roman citizenship. I could say as much of his brother, stamped as unworthy by this unlucky relationship, and incapable henceforth of being a useful member of your body."

[Interrupting shout]: "Here now, Tiberius Caesar Germanicus! It's time to let [us senators] understand what your talk is driving at - already you've reached the very limits of Narbonnese Gaul!"

There is a similar example from the reign of Marcus Aurelius. It is essentially an argument over budget regulation of public spectacles, though senators seem to have been a bit more polite:

[A senator]: Concerning the prices of gladiators, however, I expressed a little while ago an opinion in accord with the prescription of [Marcus Aurelius' speech], but I think the rule is formulated in such a way that those prices apply to those states in which prices of gladiators have been flagrantly high. In respect to states which have a rather weak commonwealth, on the other hand, I suggest that these rules which are prescribed in stronger communities be not so rigidly maintained and not place burdens upon them beyond the limit of their strength...

Whilst this is rather mundane, it was important for people to know that these new price regulations were in effect and why they had been implemented. Claudius and Marcus Aurelius were keen on political transparency, and we do not know how normal this practise of publishing unedited senate transcripts across the empire actually was. Other emperors might have just sent out a summary rather than the full debate or ruling, but it seems to have been the main way new laws were communicated.