Odds of Getting In
The job's got an eight percent growth rate, so if you're experienced and know what you're doing, there should be work for you (source).
However, a lot depends on where you live and your experience. If you have no experience and live in the boonies, you're going to have a hard time getting work.
You'll probably have better luck getting started with a big agency so long as you can pass their tests for keyboarding and editing. Some newbies may have a hard time breaking into the market though, especially in areas that are oversaturated with transcriptionists looking for work.
Doctors may think they can rely on speech recognition software to do the job, but this isn't realistic. They will still need editors to go over the material and make corrections. It might, however, lower both what you can charge and the amount of work you'll have to do.