Latin was the official language of the empire. No matter what the "common local" language was the Latin served as the international language. Much as English is the standard language for airline pilots today. It was also the language for the law.
The native language of the Romans was Latin, an Italic language the grammar of which relies little on word order, conveying meaning through a system of affixes attached to word stems.[201] Its alphabet was based on the Etruscan alphabet, which was in turn based on the Greek alphabet.[202] Although surviving Latin literature consists almost entirely of Classical Latin, an artificial and highly stylized and polished literary language from the 1st century BC, the spoken language of the Roman Empire was Vulgar Latin, which significantly differed from Classical Latin in grammar and vocabulary, and eventually in pronunciation.[203] While Latin remained the main written language of the Roman Empire, Greek came to be the language spoken by the well-educated elite, as most of the literature studied by Romans was written in Greek. In the eastern half of the Roman Empire, which later became the Byzantine Empire, Latin was never able to replace Greek, and after the death of Justinian, Greek became the official language of the Byzantine government.[204] The expansion of the Roman Empire spread Latin throughout Europe, and Vulgar Latin evolved into dialects in different locations, gradually shifting into many distinct Romance languages. (http://en.wikipedia.org/wiki/Ancient_Rome#Language)