Is it a breach of privacy to list the names of past interns and supervised students on your academic website? I've seen many personal web pages of (US/UK/EU) academics that have a list of all the PhD, MSc/BSc or internship students they've supervised before. Is this allowed or does it break privacy of the students?
The governing federal law in the USA is FERPA - the Family Educational Rights and Privacy Act. Under FERPA a school may generally disclose what is identified as "directory information," things such as names, email addresses, photographs, etc. However the student has the right to request that their directory information be excluded from public disclosure, so it's possible that they are not covered here.

Note that, as in most cases, individual states may have their own privacy statutes.

If you are really concerned then the prudent thing to do, regardless of legal jurisdiction, would be to ask the student in an email whether they are OK with the posting of their name and potentially other information. The email provides a reasonable level of proof that you did your due diligence prior to posting student information. 

US DoE guide to FERPA: https://www2.ed.gov/policy/gen/guid/fpco/ferpa/students.html