STDs are classified as reportable diseases for a reason. Because most of them can only be transmitted through sex, it should in theory be possible to stamp the treatable STDs out of existence through reporting and contact tracing. The problem is that these diseases are so often asympotmatic that it can be difficult to locate everyone who is infected and help them to find treatment.
In addition, doctors aren't always all that good at reporting diseases - even when they're legally obligated to do so. It is possible that doctors at free STD testing clinics are more used to reporting, and therefore more likely to do it correctly than doctors in private practice, but the responsibility to notify the government of these reportable diseases is the same. In theory, any STD notification sent by a free STD testing clinic would also be sent by your private physician.
In other words, you shouldn't be afraid to seek treatment at free std testing clinics because you're worried about disease reporting. If anything, the doctors who practice at free std testing clinics are likely to be more sensitive to your concerns about social stigma and other related issues. They deal with patients with STDs each and every day, and so it's highly unlikely that anything you can say will shock or surprise them. Reportable diseases are their bread and butter.
Fortunately, whether you choose to get tested and treated at one or another of your local free STD testing clinics or at your doctor's office, the disease reporting shouldn't be a burden on you. If you are uncomfortable with the idea of your identity being reported, there are generally systems in place for anonymous and confidential disease notification, because fear of identification shouldn't stop anyone from being tested. After all, the purpose of disease reporting is simply to make certain that individuals receive appropriate treatment and to track the progress of STDs in various communities. Along with other benefits, such surveillance can allow local, state, and national organizations to design more effective intervention programs that will, hopefully, help keep more people from being infected in the future.