My understanding is TB is spread by the infected person coughing into the air, and then those droplets are inhaled by surrounding people. Some will show a positive to the TB tine test, but they're not contagious. Others go on to develop their own infection. Depends on the quality of the immune system, as usual.
I've never been clear whether I actually had it or not. The test was clearly positive, done as part of routine physical. I could think back to when I was probably exposed, because I'd worked in a medical clinic for five years, for a large part of the time doing chest x-rays. And I definitely remember an episode while I worked there when I was coughing all the time, and couldn't seem to shake it. Finally, one of the doctors deigned to have a look at me, and after several breathing treatments I was fixed up. No one's ever said "you had TB," and when a later doctor mentioned the lung scars, I was taken aback and didn't think to ask, "Does that mean I had an active infection?" Maybe next time, if I have my wits about me.
If I did have an active infection, I can only pray I didn't pass it on to anybody else, especially as I sang in the church choir. :( At least my husband (then boyfriend) was cleared. :)