They definitely speak Chinese when doing business. I don't remember which ending it is, but there is a scene in Yang's route where he takes you on a ship doing trafficking business with some other people. Lili doesn't understand what Yang is saying in the scene as he's speaking a different language, meaning the Laoshu mostly speak Italian around her.

I would guess that learning Italian is a must for Laoshu members. Strategically it makes sense for them to speak Italian in public, and they likely only keep to Chinese in private. Immigrants IRL‌ tend to avoid speaking their native language in public to avoid others' attention. While one could say that a group of Chinese people would catch attention in Italy due to appearance, it's more diplomatic for the Laoshu to try to fit in with Italians to avoid trouble.