At least this is the latest position in Italy, which is rather interesting, and provides some lead in controlling this pandemic in the workplace, reducing the risk on rights and freedoms of employees. The relevant paragraph from the article worth reading and I am referring to is quoted below.
The Italian data protection authority held that serological tests run on employees are privacy compliant in Italy provided that the occupational doctor is the data controller and is the sole individuals aware of the results of the test, communicating to the employer only the suitability/unsuitability of the employee to perform the working activity.