The U.S. is getting a first glimpse of what it's like to experience COVID-19 outbreaks during this new phase of living with the virus, and the roster of the newly infected is studded with stars.

Cabinet members, House Speaker Nancy Pelosi, Broadway actors and the governors of New Jersey and Connecticut have all tested positive. Outbreaks at Georgetown University and Johns Hopkins University are bringing back mask requirements to those campuses as officials seek out quarantine space.

The known infections likely reveal only the tip of the iceberg — with actors and politicians regularly tested at...