Simple. The system doesn't track every person you're in contact with.
A simple method is to have phones directly exchange random numbers over bluetooth. If and only if someone tests positive, they publish their phone's recent random number transmissions. All phones download all published random numbers (just random numbers, no personal info) and see whether they've received any of them via bluetooth.
This way the phone knows whether it's been in contact with someone who tested positive, but nobody else knows. Since the user has a strong incentive to get a covid test at that point, that's fine.
A simple method is to have phones directly exchange random numbers over bluetooth. If and only if someone tests positive, they publish their phone's recent random number transmissions. All phones download all published random numbers (just random numbers, no personal info) and see whether they've received any of them via bluetooth.
This way the phone knows whether it's been in contact with someone who tested positive, but nobody else knows. Since the user has a strong incentive to get a covid test at that point, that's fine.