What kind of privacy policy should we implement for European customers?

Hello! We are a startup based in New York developing a software product for the Alternative Energy market. We have had a successful initial launch for our MVP and there are some customers, especially from the US, but also from the UK, Belgium and Switzerland interested in testing our products. Our question is regarding customer data: What kind of privacy framework do we need to comply with for our European customers? Would this be regarding the new EU Privacy Shield? What about Switzerland?