A highly motivated student will progress faster without a "teacher" than a poorly motivated student that has one.
In most cases (or at least many of them), having a teacher is probably best. Why try and reinvent the wheel? You simply should be able to learn faster and better with having someone show you how to do it correctly from the start.
Still, not everyone has a teacher yet many seem to progress nicely.
I think it may be more about motivation. Many people find ways to teach themselves to play piano and have done so forever. Some of the greatest musicians are self-taught.
Maybe it's natural talent or possibly relentless motivation but no matter the size of the container, the cream will always rise to the top.
People who are highly motivated to learn a task usually will. However they go about it, it WILL get done.
Many of us here on this forum are self taught, or worse! (no names!) and while we may not be the best, we proudly hold our own. Proud of what we have accomplished all on our own. A personal achievement of sorts.
Your succsess will be almost entirely dependant upon the amount of your desire to learn. The stronger your motivation, the better you will probably become.
Of course, as always, I am making all of this up. I'm probably right though!
I think that having a teacher can be a great motivator, but that isn't always the case. There has to be some inward motivation on the part of the student to succeed and take instruction for it to work. It's teamwork, really. The biggest advantages of having a teacher:
-Material will be presented in an orderly manner. A teacher knows what a students needs to learn next and knows the appropriate repertoire that will help them get there.
-They can also make sure that technical issues are worked out to prevent injury, pain, and poor musical execution of a passage or piece.
-Having a pair of ears listen to your playing to determine how well you're doing or what you may have missed or what might need work is invaluable feedback
-A teacher can give you practice suggestions on how to work certain difficult passages and correct errors. There are so many different ways to practice, that learning these techniques actually makes practicing even more fun, and it will get you results quicker than if you simply keep playing through the passage over and over again.
Having said that, of course someone can learn without a teacher. People do it all the time, and if you are at a point in your life where you can't afford it, then you make the best with what you have. I encourage you to save up or plan in the future to have some lessons, even if it ends up being every other week for a few months. I personally still alternate between taking lessons and working on my own. As a teacher myself, I still need a pair of ears to listen and offer feedback and suggestions.