Privacy regulations (HIPAA) may forbid them from saying "Hey we think you might have COVID" on your voicemail, unless you've already filled out paperwork saying "It's okay to leave a message on my voicemail with detailed health information."

Your doctor's office can do it because you probably have an existing patient relationship, and you already filled out that paperwork on your first visit.

But in the situation of a cold-calling contact tracer, you haven't signed anything at all, so all they can say is "This is the health department and we want to talk to you."