There are a few immortals like Hoffmann who did not study formally for long and went on to careers and greatness. For most, working with a teacher will help them progress immeasurably. The big difference is discipline. When working alone, sloppiness, bad habits, and a tolerance of same can creep in quickly. When working with a teacher, especially in the younger years, steps are taken to ensure a balanced approach to technique, musicianship, and repertoire. In the case of serious students, there are specific goals to be met--auditions, recitals, competitions, etc. Social pianists are given more modest goals, but goals nonetheless.
The teacher is an astute observer and listener who explains, critiques, demonstrates and inspires (and sometimes demands!). The student, therefore, works harder. In the case of a very advanced pianist, the student doesn't need a teacher as much as a piano coach to concentrate on the finest points of interpretation. By the way, because learning is a lifelong pursuit, it might surprise you to know that many teachers (perhaps your own) often have teachers. That is, they continue to work with distinguished faculty at conservatories and university schools of music or performing artists. They understand better than most how valuable it is to have a critic and mentor in one's corner--whether or not one is fully capable of studying and practicing alone.