The problem is that the tests and testing and the definition of symptomatic are changing with time and whoever is doing a study. The news rarely if ever defines how these numbers are obtained.
"Testing positive" generally means they swabbed the back of someone's nose and viral RNA could be detected on the swab. And they are likely to be infectious. Symptomatic means someone reports one or more signs from a defined list of symptoms. In the Medscape link below, there is another link to a Nature Medicine paper in which their list of symptoms did not include covid toes or loss of sense of smell because the study was done in China early in the epidemic before those symptoms were known. So asymptomatic would be overestimated.
Then there is the problem of asymptomatic versus presymptomatic. If someone in your state is covid positive, do they come back two weeks later and repeat the swab test and ask about symptoms to see if they now have symptoms?
In the Nature paper the Chinese researchers had a captive audience so to speak since the patients were in a "central quarantine" so they could ask that and several other interesting questions. They also tried to answer your questions about follow up antibody tests. Interestingly the numbers of antibodies of two types fell pretty fast, faster for asymptomatic individuals than symptomatic. Have a look at those papers, pretty interesting, and like most research, raise the need for further studies. The Nature paper includes data to indicate asymptomatic people can transmit the virus to others. And throws a caution flag for the assumption that presence of antibodies now will provide protection in 6 months. It might not be that simple.
https://www.medscape.com/viewarticle/932671
