I'm going to preface this by saying I'm not a historian; I'm a composer that has a strong interest in music history.

There was definitely a strong musical culture in ancient Rome. They had very strong influences from Greek musical culture, as Greek musicians traveled to Italy in huge numbers during the second century B.C.E., due to the destruction of Corinth and the conquest of Macedonia. These Greek musicians organized guilds, and they had enough numbers to organize public festivals for music and theatre. This importation of Greek musical culture had a huge impact on the musical culture in Rome.

There were definitely musicians that were relatively well-known by their contemporaries. Virtuosos are mentioned often by writers from ancient Rome. There were senators and even emperors (such as Caligula, Nero, and Hadrian) that were so inspired by these famous performers that they became amateur musician's themselves and even pursued instruction from their favorite performers. Nero was actually so confident in his musical abilities that he participated in musical competitions against famous performers. Some named performers are Terpnus (who Nero studied with while he was emperor), Diodorus, and Mesomedes, who was also well known as a composer.

There were many concerts in Roman life. As I stated earlier, there were massive festivals that were put on by Greek musicians that had moved to the Roman Empire. There were also contests that were frequently held that were established by Nero, who even participated in them. Roman writers mention that famous performers could be payed massive fees for singing Greek hymns and solos from theatrical works in a concert setting. Audiences were known to praise skilled playing and criticize inaccurate and messy playing. Large and extravagant concerts declined as the Roman economy declined in the third and fourth centuries, however.

Romans clearly loved music, and had a huge musical culture (even if it mostly drew from Greek musical culture). There were definitely performers and composers with fame in the Roman Empire. Because there wasn't any way to record music for playback, concerts were the only way for Romans to witness their favorite performers play. Concerts were huge in Rome, at least while the empire was thriving. While all of this isn't perfectly analogous to our modern musical culture, they definitely had their own musical culture, and it was pretty important in Rome.

Sources:

A History of Western Music, 9th Edition by J. Peter Burkholder, Donald Jay Grout, and Claude V. Palisca

"Rome" on Grove Music Online/Oxford Music Online by Günter Fleischhauer