Architects were responsible usually for both the design and the execution (management) of monumental building in Roman times. They were nearly always working under the commission of a patron (either one of the elite, or else a king or emperor). The architect was almost always a Roman or a Greek person, though the distinction is pretty fluid. For instance, two famous examples:

Decimus Cossutius was an architect who worked for Antiochus IV Epiphanes, the Hellenistic Seleucid dynast of Syria. He was a Roman, probably from Campania, as confirmed by an inscription about him found in Greece and still extant. We know nothing about the who/what/when/where/how of his training in this profession, but after him we see several Cossutii in the same business, and we often see "skill" positions like this handed down through families. Antiochus hired him in the 160s BCE to finish the long-abandoned great temple of Zeus at Athens, and this was one of the most prestigious architectural tasks of the day.

Hermodorus of Salamis was a Greek, from the western isles, and he did architecture at Rome. Around 143 BCE, the famous triumphal general Q. Metellus Macedonicus hired him to build the Temple of Jupiter Stator in Rome. It was the first fully marble building in the capital, made possible by Rome's recent...acquirement of the best marble quarries in the eastern Mediterranean. Metellus competed with other prominent Roman aristocrats for the services of the Greek person, and the construction of buildings for the beautification of the city was an important route through which Roman elites displayed their importance and prestige. The Greek architect's skill with marble, and his reputation, no doubt made him highly desirable for his services.

We know of both of these guys through ancient sources: the Roman author-architect Vitruvius, and some other, later, historians like Velleius Paterculus. Sometimes we find their names recorded on inscriptions placed on the buildings they designed or built, along with their patrons.

When the system of emperors took over, they became (obviously) the most important architectural patron, and the majority of the major building programs in the capital stem ultimately from them from that point onward. Private commissions still persisted, however, even into the late antique period, where we find the Church increasingly becoming a patron of civic construction. Beyond the capital and out in the provinces, large-scale building was often still initiated by private individuals, but also sometimes by civic collectives in competition with neighboring cities. In their eagerness to out-do neighboring towns and impress the emperor back in Rome, these cities often ruined themselves financially. Pliny the Younger, while governor in Bithynia, describes how destructive this "building competition" could be. Un-needed aqueducts, temples, and sports arenas were common culprits (compare American professional football stadiums today).

As I mentioned above, we know virtually nothing about how architects were trained, or how engineering know-how was transmitted between individuals or generations. In 55 BCE, Caesar ordered his legions to bridge the Rhine, which they did without any apparent trouble. Clearly someone with engineering know-how was there with the army, but we don't know how exactly that worked. I talked about this a lot more here.

For a great write-up of this topic (including about Cossutius, Hermodorus, and others), see Anderson's chapter "Architect and Patron" in the excellent A Companion to Roman Architecture, ed. Ulrich and Quenemoen (Blackwell 2014).