Should private businesses have the right to ask customers for their vaccination status?

In an effort to end the COVID19 pandemic many governments imposed vaccine mandates on people who wish to enter private businesses. Politicians who supported the mandates argued that it would stop the spread of CV19 and encourage people to get vaccinated against it. Opponents argue that vaccination status is private health information and people should not be forced to share it. Proponents argue that unvaccinated individuals are responsible for spreading CV19 and prolonging the pandemic.