If you test on antigen that matches the virus, then that will return a negative result in the very beginning of infection and after recovery.

If you want accurate numbers for statistics, including people who never had symptoms, you can do antibody testing on a random sample of population, that will indicate people who had the disease - but that's kind of too late for that particular location and those particular patients, that's for epidemiological studies and helping other locations. I recall reading about one such study for Covid, but I don't remember which country did it.

I heard in an interview with Michael Osterholm, that scientists in China started testing people who had come in contact to those who were sick in the early days of the outbreak, and found people who were not showing any symptoms, but had a large amount of viral load in their throats they were actively breathing out.

What kind of testing they do in those situations?

The first one - if you have the virus, it's multiplying in you and you're infectious then that test will detect it (during that time) no matter if you have severe symptoms or light symptoms or if for you the infection doesn't have any symptoms that you would notice.