I feel like I’m missing a reference here, but are you ok?

That’s not the difference. Both words have noun and verb forms.

Immigrate = to move to a place

Emigrate = to move from a place

Immigrant = a person who moved to a place

Emigrant = a person who moved from a place

So they would be emigrating from the US and immigrating to Italy. They would be a US emigrant and an Italian immigrant.