Of course, today's world offers more opportunity to encounter speaker of the same language from different nations. But I reckon educated people of the time were also in contact with other nations' inhabitants who spoke the same language?
Sure. Here's a couple examples:
Catullus #84 is about a man named Arrian who added an "H" sound to his words to sound more Greek (since speaking Greek was amark of a well-educated upper-class Roman. Catullus loathes this and mocks him savagely:
"Hadvantages" Arrius was saying whenever he wished to say advantages And ambush he was saying "hambush," And then he was hoping that he had spoken wonderfully Under the circumstances when he said "hambush" as much as he was able, I believe, thus his mother, thus his free uncle, Thus his maternal grandfather and grandmother had spoken. With this man having been sent into Syria, the ears of all had rested: They were hearing the same thing more softly and more lightly, Nor afterwards were they themselves fearing such words, When suddenly the horrible message is brought that: The Ionian waves, afterwards Arrius had gone there, Were now no longer Ionian but "Hionian."
(translation from Wikisource)
Another example, Cicero advises in De Oratore to speak Latin with (unsurprisingly) a Roman rather than a provincial accent:
Consequently as there is a particular accent peculiar to the Roman race and to our city, involving no possibility of stumbling or causing offense or unpleasantness or objection, no note or flavor of provincialism, let us make this accent our model, and learn to avoid not only rustic roughness but also provincial solecisms.
A professional classicist may be able to go into greater depth, but that's what I could pull up. Unsurprisingly, Romans, it seems, like people throughout history, mocked the accents of speakers of their language from elsewhere.
The emperor Vespasian (AD 9-79) was Sabine in origin, from reate, some eighty kilometers north east of Rome (he also spent time as a child at Cosa, some 140 kilometers north of Rome), and was noted for his country accent. Suetonius tells the story of his encounter with an ex-consul called Mestrius Florus who corrected his pronunciation of plaustra (cart) rather than plostra. Vespasian greeted him the next day as "Mestrius Flaurus". (Suetonius, Lives of the Caesars, the Deified Vespasian, XXII)
He would have meant it as a joke. Vespasian, unlike Cicero, was a whole lot more easy going about most things in life, his rural origins not least of all. He was, by the way, like Cicero a novus vir, a new man, a provincial (Cicero was from Arpinum, a hundred kilometers south of Rome) who rose on money and merit.