Late January, the U.S. Department of Commerce published a notice of proposed rulemaking for establishing new requirements for Infrastructure as a Service providers (IaaS) . The proposal boils down to a 'Know Your Customer' regime for companies operating cloud services, with the goal of countering the activities of "foreign malicious actors." Yet, despite an overseas focus, Americans won't be able to avoid the proposal's requirements, which covers CDNs, virtual private servers, proxies, and domain name resolution services, among others.
Generally yes, it would matter a lot how it was structured. Today you couldn't call up AWS and ask for the details on a service owner out of privacy reasons and there are ways to register things by proxy. If they started stripping those kind of protections away though there's bound to be some pushback.