For children in public school, success can be measured at each pop quiz and chapter test. If a student can pass a test after the teacher has taught the material, then all is well. Of course there is some debate in the success of teaching to the test, a teacher who can promote a class full of children who are ready for the next grade level is typically successful.
The success of homeschoolers is a little harder to measure. In my state we only take standardized tests once every three years, so there is a periodic pat on the back when the kids are successful. As homeschooling parents who get to see each lightbulb, or aha moment, there are also successes. But at what point are we successful to the outside world? Some people want to mandate testing to all homeschoolers so they can prove children are being taught. The problem with that is homeschoolers will be tied to a school curriculum plan for that to happen and most of the joy will be sucked out of homeschooler as more parents begin teaching for the test.
A pretty good measure of homeschooling success is the ability to thrive at a college level, meaning elementary, middle, and high school homeschooling was successful. As a homeschooler, you can opt to actually take college level classes to prove that success to yourself, your children, and to prospective colleges.
With a child who has taken his first college level program examination, or CLEP exam, I am certainly feeling vindicated and secure that I am doing the right thing as my son was able to study for and pass a test for a course that I would have been hesitant to take.
This is what success looks like... the ability to follow a study guide and to prepare themselves for a college level exam meaning they have been given enough skills to make you an independent learner.