There were no newspapers, public or private, in the Roman world. The spread of information in the Roman world, to the extent that it was done through writing, was largely accomplished through posted notices. The most important of these, at least at Rome, were the alba, boards on which official notices were posted publicly within the city. Probably the best known of these is the praetorian album, on which the notices of the praetor were posted, but there were many different kinds posted by many different magistracies and carrying various sorts of official public information. These alba were generally lists, though, and often of a legal character, and you wouldn't find out what was going on in some faraway province by consulting them, though you could find the latest rulings of the law courts and the latest leges. Within the city information would also be spread orally, through praecones (basically criers) and other minor officials employed by magistrates, or occasionally by magistrates themselves at public assemblies of various kinds. Again, though, this was rarely "news" from throughout the empire, but really local (often legal) information and instruction. There was the daily publication of the acta senatus and the acta populi, the daily "doings" of the senate and the people respectively. This was introduced in 59 during Caesar's consulship, although Augustus forbade the acta senatus from being published. This was probably the closest thing to a newspaper in the city, being published daily and posted on tablets just like the rest of the alba, but it wasn't really like a newspaper as we know it. These were official notices, and they didn't really include what a modern paper would--Petronius includes a comic version of the acta, for Trimalchio's household, of which the following is an excerpt:

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 says that it continued with a list of aediles' edicts, wills that Trimalchio was excluded from, the punishment of some of Trimalchio's slaves, etc. This is of course a parody, but it's easy to see that the acta weren't exactly the New York Times.

The acta were read in the provinces apparently, but there seems to have been no official system of actually distributing them. Official information was passed when possible by state officials, including the viatores and statores, minor officials whose duties might include acting as messengers. Even official information, however, was generally spread by private citizens (at least in the early Principate and the Republic--during the Principate a courier system for state documents began to develop), either the slaves or dependents of various magistrates or publicans and other tradesmen. For private correspondence and information the situation was not too different. The Roman state at no time had a postal service, and private citizens entrusted their letters to slaves and dependents or sometimes even to passersby and merchants--even Cicero talks about handing off letters to travelers and his friends' slaves. Of course, the army and other particular state functions had unusual access to information (the army, after all, or at least magistrates acting through the army, had the exclusive ability to send information by warship or military march), but of course the vast majority of people would have no access or interest in the information thus conveyed.

Mostly, however, information and news was spread orally. Obviously praecones and public speeches disseminated news orally, but what I mean really is by rumor. Ray Laurence had a good article a few years ago about rumor's involvement in politics in the city, and it's quite clear that a lot of information (political or otherwise) was spread by word of mouth. Candidates might have political pamphlets handed out or their supporters might paint endorsements on the walls or distribute objects through the city (those are bowls that Catiline used to support his candidacy), and magistrates and praecones might deliver public addresses, but really the access of the individual Roman to these means was pretty limited. Even within the city the average day-working laborer did not have the ability to miss a day's pay to go listen to a speech in a contio, and access became even more problematic as the provinces became more "Roman" and connected during the Principate, thus desiring the dissemination of more information at levels lower than provincial magistracy. Information was therefore spread orally, as people played a giant game of telephone throughout the city and eventually through the provinces. Precious information was made available to a certain few--those who had heard a magistrate speak in a contio, or had received a letter from the provinces talking about some event there, or had actually bothered to read what was on the acta and alba, or had been told by their patron that morning about this or that military victory or law to be brought to the assembly--and from there it was spread throughout the city (or the provinces, if we reverse this).

There was no newspaper, since newspapers require a degree of mass literacy which was more than likely absent in the ancient world, and an inexpensive means of production. With no printing press or paper, there was no way a newspaper could be economically viable. News was spread largely by word of mouth, and in a variety of way. First, the men at the top of society kept in touch with each other through letters, and these usually passed along news along with other information of more personal interest. The best and most extensive example of this sort of thing is the collection of Cicero's letters. But other collections of letters exist from other time periods (Pliny for the early empire, Symmachus for the late) and show how letters could pass along news. Presumably merchants and other literate men from outside the very upper echelons of society kept each other informed through similar epistolary networks.

Official news was passed along through ambassadors and letters as well. When a city wanted to request something of the emperor, they sent a citizen to ask him. The emperor then sent the ambassador back with a letter giving his response. We know of these because when the response was favorable the city often had the letter inscribed. At a more regional level, cities sent ambassadors to each other to announce local news all the time. There was constant stream of emissaries flowing back and forth announcing upcoming festivals or making requests for this or that. Once more, we know about these because cities often set up inscriptions honoring the men who had taken the time and trouble to go on these embassies.

Finally, non-written communication, particularly through coins and monuments, was probably the best way to ensure a wide group of (often illiterate) people knew about something. So when Germanicus (the nephew of the emperor Tiberius) defeated the Germans in AD 19 and recaptured some of the legionary standards lost in the Battle of Teutoberg Forest in AD 9, coins were issued featuring Germanicus and the inscription "the Germans have been defeated, the standards of have been returned."