PE Requirements
There are specific Coast Guard PE requirements, though most yacht designers won't find them - in general if an analysis would require a PE on land, Coast Guard has policy that it is expected on a ship. This might be an ASME B31.1 analysis or similar things.
However, the requirement for PEs is mainly up to the states, under the separation of power clause and subsequent Supreme Court rulings regarding PEs.
If an analysis is legally required to assure public safety, and is described as, or otherwise requires engineering analysis, a PE may be required, though there are numerous exemptions and circumstances. The most obvious example is preparing stability and trim information as required by 46 CFR 28.500 for commercial fishing vessels over 79 feet long, which is legally required and is specifically described as requiring "naval architecture calculations". There are a number of other analyses for small passenger vessels carrying more than 49 passengers and so on that might also meet these requirements, (in the absence of industrial exemption and so on). However, most of these analyses involve topics, such as damage stability, that yacht designers don't get involved in anyway.
Note also that there is no "push" to require licensure. The laws on engineering licensure go back mostly to before WWII, and if someone had a complaint, the laws were enforced. As a result of various circumstances, most involving issues other than legal ones (some military positions require PEs, for example), a number of SNAME members asked that SNAME help them to get an exam that would allow them to obtain a PE (which is a general license) by taking an exam in NAME instead of ME, EE or civil. SNAME and NCEES developed this exam, and that is all. Anyone who needed a PE before needs one now, anyone who didn't doesn't.
Mr. Gerr is correct that there is a lot of poorly informed opinion on this, but as long as yachts and many yacht-like commercial craft don't have any legally required engineering analyses, a PE is not required to perform these analyses, as long as the practioner doesn't imply that what they are doing is engineering, not design.