Cities that offer good transit service don't have to worry about the competition. Those transit services already offer fast and reliable service at a reasonable price.
On the other hand, cities that offer horrible transit service need the competition. They need to realize that poor coverage, poor scheduling, unreliable service, and drivers with poor safety records are unacceptable. If they don't realize that it is unacceptable, then maybe they should shutter their doors and let the private sector take over. (This coming from someone who normally supports a strong public sector.)
To give you an idea of what I mean: I work two jobs in a city with poor transit service, so I decided to sit down and do some math one day. The end result is that taking the bus cost significantly more than taking a taxi. That's a single person in a regulated cab, and not the shared-ride service mentioned here. Yes, a great part of the cost was from lost income. Yet it was real lost income in my case because I had to negotiate my work hours around transit. For other people, the loss of income will come in other forms: being unable to accept a job due to transit coverage or scheduling, or losing a job because unreliable service results in an unreliable employee. For other people it will result in a diminished quality of life, simply because much of their time is spent waiting for or being in transit.
(To give you an idea of how inefficient transit is in my city: if it takes 30 minutes to walk somewhere, you may as well walk since the bus is going to take longer. If you have to be somewhere at a particular time, you can usually increase that 30 minute walking radius to 1 hour because that bus that "arrives 10 minutes early" will end up arriving 10 minutes late so frequently that you will end up unemployed.)