Probably not as much as you would think.

First we have to talk about what we mean by the Fall of Rome. Commonly the abdication of Romulus Augustulus in 476 is used as the definitive date as the Fall of Rome, but everyday Romans living their everyday 5th Century lives wouldn't have noticed much difference between 475 and 477. For at least a century there had been a pattern of having two coequal Emperors, one in Rome (then in Ravenna) and one in Constantinople. At times, e.g. the reign of Theodosius, there was just one Emperor. To the people of Rome, 476 felt like such a time; rather than two Emperors, Romulus Augustulus and Zeno, now there was just Zeno.

It's also important to consider the manner in which the German Tribes invaded the Roman Empire. There's a popular belief that they came in and tore the whole thing down. Although there are examples of German Tribes attacking just for plunder, e.g. the 410 Sack of Rome, by and large they weren't seeking much more than a homeland for their people. Different groups effected this in different ways. The Franks set up a government system that was meant to function side by side with the Roman system, the Visigoths asked for the right to farm 1/3 of the Roman landholders' land, and the Ostrogoths established their kingdom as a vassal of the Eastern Roman (i.e. Byzantine) Empire, continuing to apply most of the old Roman systems of law and administration.

So, to answer your question, unless you were unlucky enough to be involved in one of the Sacks, you wouldn't see mass execution or mass exodus with the fall of the Western Roman Empire. You might have some new neighbors whose Latin wasn't great, but you otherwise wouldn't notice much difference.

Source: The Invasion of Europe by the Barbarians, JB Bury Byzantium : The Early Centuries, JJ Norwich