The privacy act is very clear on when and how private information can be disclosed and it is no different with Covid 19 – see following link https://privacy.org.nz/blog/faqs-on-privacy-and-covid-19/.

In short you need to gain the consent of the affected employee to have their identity as a sufferer/person in isolation disclosed.

However, if there is any concern that others in the workplace may have been exposed, they will need to be made aware so that they can be tested and go into (further) isolation. This doesn’t necessarily mean it is necessary to identify the employee in question but obviously given some circumstances (e.g. small office) this may be unavoidable.

One of the exceptions is where disclosure is to prevent or lessen risks associated to health and safety.