Barely a week passes without someone complaining about the teaching of English or foreign languages, usually because it involves too much, or too little, grammar. The ancients also had to face the problem. Clearly, non-Romans who wanted a career in Roman high society, the courts, civil administration or the army needed to learn Latin. So they did, and by the 2nd century AD, the Greek essayist Plutarch was able to say that almost all men used Latin. Certainly, as the Vindolanda tablets demonstrate, the Latin of the Germanic officer Cerealis was very respectable.
But Romans also admired Greek culture enormously, and Latin literature drank deeply at its well (the statesman Cicero could switch effortlessly between Latin and Greek). Trade too provided incentives for Romans to learn Greek; and as it was the lingua franca of the Mediterranean, and there were huge numbers of Greek slaves in Rome as well as immigrants, more Greek was probably spoken in Rome than the local lingo.