Can someone who has been given a vaccine "infect" another person with the virus used in that vaccine? For example, let's say Bill is vaccinated against tuberculosis. If Frank is exposed to Bill, is there a chance he could "catch" the tuberculosis vaccine, and become immune?
Most immunizations are not transmissible. A few are. To be transmissible, an immunization would have to be a live attenuated vaccine (i.e., a vaccine that can reproduce in the host, but doesn't cause disease), AND have the necessary reproduction rate and virulence factors that would allow for person to person transmission. There are many live attenuated vaccines in use today, but only one that is transmissible, Oral Poliovirus Vaccine, or OPV. OPV is currently not in use in the US, but is used in the developing world, and can circulate in communities. Some would argue that this is a good thing, as it further disseminates immune protection to the population, but it is a little problematic, because it is not without risk, including reversion to a virulent type that can cause Vaccine associated Paralytic Poliomyelitis.
On editing your question, I'm reminded that your example was of TB. I would note that BCG, the vaccination against Tuberculosis, may be a particular case, and I'm not as familiar with it except as a complicating factor in interpreting the results of a TB test in U.S. immigrants. BCG is a live attenuated strain of Mycobacterium bovis, can (rarely) produce disseminated disease itself, and the mycobacterium itself can be isolated in the blood of individuals some time after vaccination. There is also a possible report of vertical transmission of the mycobacterium (from mother to child), but these results use novel methods, and I'm unfamiliar with this group's work. Whether this would lead to immunity of the child is definitely an open question. BCG even when administered in the standard way (as in intradermal inoculation) isn't particularly good at producing immunity, though it is good at preventing meningitis and miliary TB in young children. I'm not aware of any documented case of horizontal person-to-person transmission, but wouldn't declare it impossible.