Contact tracing to get the right people tested should be done for the testing to be effective and efficient. It's entirely possible to do it without violating anybody's privacy. Below, I will give the specs of a system that does it.
It's done using a mobile app that relies on Bluetooth to sense proximity.
Each app instance of the app calculates a pseudonym with a hash function (one-way encryption) form the device ID and the identity of its owner. The identity behind the pseudonym is not stored anywhere but the owner's device. There is no database connecting the pseudonyms to real world identities anywhere.
Each app instance stores only its own contact data containing the pseudonyms and the time of being in proximity to them. The data are deleted when they become epidemiologically irrelevant.
When a user tests or is otherwise diagnosed positive for COVID-19, he/she gets a code from the healthcare provider that tested him/her that can be used once that is required for broadcasting the information that their pseudonym has tested positive for COVID-19. Those whose recent proximity data contains that pseudonym self-isolate and contact a healthcare practitioner to get tested.
There is no central database of anyone's contacts anywhere. The proximity data is stored locally on the app users' devices. The users are anonymous to each other and the authorities. Not even the test administerers are told the pseudonyms of those tested. They just provide the app user a code (that can be used only once) that enables them to report their positive COVID-19 status. The code is needed to prevent any malicious app user from interfering the network by reporting false positives.
The positive results could be reported using the Tor network help keep the IP addresses of those devices whose owners are COVID-19 positive hidden.
Under this system, nobody could be forced to take a test after detecting that a pseudonym that was found positive for COVID-19 appears in their recent close contacts. Some may choose to not get tested. Some people may not even want to use the app despite its strong privacy protection.
But it is not necessary diagnose and quarantine every carrier to control the epidemic. It is sufficient to keep the reproductive number R0, or the number of people the average carrier infects below one to suppress the epidemic. And keeping it above but close to one, which is what is taking place in most countries that are practicing social distancing now is sufficient to slow the spread of the virus down enough to ensure the ICU capacity to be sufficient. The problem is lockdowns wreck economies. The only way out is to use whatever means necessary to prevent transmissions while allowing the economy to function. Automatic contact tracing combined with testing is one of the best ways to accomplish that. And good news are it's possible to do that without sacrificing privacy.