The western part and especially Rome itself had already suffered large population drops thanks to pandemics long before the Barbarian invasions. As far as I know, not much changed at start, trade continued and things seemed normal. "Barbarian" influences in the Italian language is really limited, which suggest a quick assimilation of the Barbarians, at least in Italy.

I am by NO means an expert, the little knowledge I posses comes mostly from a teaching company lecture-set about the early dark ages.