It is highly unlikely that chlamydia or gonorrhea will be eradicated without treatment. Although the symptoms often subside, the bacterial infection is usually still present. It can spread to other parts of the reproductive system, causing serious complications (e.g. pelvic inflammatory disease in females and epididymitis in males), and it can still be transmitted to any sexual partners.
Do not take the risk and get tested if you have been exposed and/or are displaying symptoms. Treatment is simple and effective.