How do you prove that people have been vaccinated without putting their privacy at risk? The technology and best practices to make it happen exist. It's far from clear, however, if they're being used.
The government should be explicit about what proportion of the population will need to be vaccinated to warrant border reopening. Australians could then measure progress towards that goal.
If a trans-Tasman travel bubble were to be established, passengers would likely need to use ‘vaccine passports’ to prove their vaccination status. But any tech-based system comes with security risks.