There are ups and downs to state-run or privately operated. London Underground used to be government operated, but it was very dirty and didn't run on time. However it was super cheap because profits were of no concern.
When it got privatised, it became much cleaner because people were actually paying proper money for it and they had a need to please travellers so they'd continue using it. And since its privately run, they have to pay the workers out of the profits alone so prices keep going higher and tube drivers keep going on strike to get more money, so we have to keep paying more to use them.
New York City Transit is operated by the MTA which owns transportation around New York. MTA runs the Long Island RR, Metro-North RR(Connecticut and lower New York state), Long Island Bus, NYCT, MTA Bus, and bridges and tunnels around New York.
http://www.mta.info
Weird, I know more about the city you live in than you do.