Since it is the public who uses and has the need for public transportation, it is they who should own and be in control of its use. A corporate owner would likely seek to maximize their profit which could come at the expense of the service needed by the public who are using it. To ensure their needs are met and maximized, the public should be the owner.
Public transportation should be owned by the public, but it should be run like a private business. Many cities public transportation systems are over budget and run badly because the officials that run the systems are not concerned with going over budget. The public should own them and be able to decide who operates the system.
I was once in Buenos Aires, Argentina with my girlfriend who happens to be from there. We approached the train station, walked through the gates, and I asked if we have to pay the conductor. No, she said, the train is free for everyone to use. Just walk on. That's amazing and necessary, especially in big cities.
We have a very good public transport system, and it could be improved better, but I don't believe it should be owned by the public. The government or state knows better how to improve the public transport system we have in place, plus its interest is not on money but rather helping the communities and people who have a low income.
Public transportation should be owned and operated by the government and not the people. It does not make sense for there to be multiple companies providing public transportation. There is no need for competition, because there should only be a single system that is too big to fail, and the prices of which do not fluctuate because of the market.