I know businesses would have had signs and advertising up outside their premises, but would we see large spaces dedicated solely to advertising things other than nearby businesses? Were there businesspeople who made significant money off of advertising?
Before all else, we need to understand the political circumstances that allowed for the rise of advertisement in Rome.
In De Officiis, Cicero stated that the majority of occupations that weren't the owning of an agricultural estate were less than desirable, considering that trade on a small scale involved contact with the masses, making it an undesirable job. According to Evan Sage in Advertising among the Romans, this conception, paired with the barring of Senators from "engaging in trade at all", made business a less important element of everyday life than it is to us today.
In 59 b.C., Caesar established that Senate activities were to be made public via reports, in a rather cunning example of an early form of Realpolitik, with the intent of diminishing the Senate's power. This reports were called actas, and began to be the norm in massive communication of events to the populace, particularly after the fall of the Republic. One of the effects the weakening of the Senate had, was the gradual social acceptance of trade as a whole, because as Senators, the once most revered figures in Roman culture, stopped being as influential, their views and opinions against trade and business as a whole began to have less of an impact, turning trading into a more acceptable occupation.
Actas were distributed throughout the provinces as a form of political communication. However, as Sage further states, the commercial application of such practices would have been highly prohibited due to costs in the production of actas.
Regarding the specific usage of signs and advertisements, August Mau explains in Pompeii: its Life and Art that in Pompeii, signage of specific shops and establishments was a fairly commonplace practice, with several real estate signs describing houses and businesses for rent. Picking up this line of thought, professor Fred Beard in The Ancient History of Advertising: Insights and Implications for Practitioners: What Today's Advertisers and Marketers Can Learn from Their Predecessors explains that in Rome, as in Pompeii, signs were positioned in streets near shops to advertise their products, as well as to make the population aware of upcoming events.
So, to summarize and finally answer your questions: even though public-oriented advertisements existed in Rome, it wasn't a wide practice. You would've found several, scattered signs dedicated to specific products, services, events or businesses, instead of a vast amount of them, or large spaces solely dedicated to it. As as a result, advertising wasn't viewed as a business per se, but rather an element to business as a whole.