Ok, first, if you have a positive TB skin test you have definitely been infected with the bacteria. It is a common misconception that positive means exposure only. There is no way for your body to produce an immune response, and have a positive skin test, unless the bacteria comes in contact with your immune system. That means the bacteria is in your body, somewhere, most likely the lungs. BUT, let me repeat, BUT, that does not mean you have active disease now or maybe ever.
This is the reason you can be TB positive without being infectious. Your immune system can do wonders in walling the organism off. A positive skin test, at least in the U.S. , means treatment with at least two antibiotics for a period of 3-6 months, maybe longer if you are immunocompromised. Treatment is necessary to hopefully prevent the infection from becoming active at a future point in time, even if you are not infectious.
There is also the possibility that a positive skin test is due to active disease. In that case you will be potentially releasing tuberculosis bacteria when you cough and sneeze. This is a more urgent medical condition, for the patient and the people that have spent time with the person infected.
By the way. Having a positive skin test does mean antibodies have been produced. There is no reliable scientific evidence that these antibodies prevent future re-infection.
Hope this helps.
Microbiologist, Public Health
University Microbiology Lab Instructor