This page lists a number of scholarly resources on the subject. However, it's very difficult to compare, because homeschools are not even close to standardized. Homeschoolers often dominate academic events like spelling bees, sports like figure skating (Michelle Kwan), entrepreneurial ventures (Andrew Carnegie), and other fields (Thomas Edison). They have better average scores than public schoolers on standardized tests. Homeschoolers who want to go to college generally excel there as well. On the social front, homeschool graduates have much higher rates of involvement in charities and community projects. Study after study confirms this. Yes, those studies don't separate "professional" homeschoolers from others, but the vast majority of homeschoolers are not professionals.
On the other hand, we have Lindsay Lohan and Adam Lanza. Although in Adam Lanza's case, he was previously in public school for many years and they couldn't handle him either. Most of the worst-case examples of bad homeschooling outcomes would probably have been as bad or worse at school. Same goes for the reverse. How can you know a particular child who did well in school wouldn't do even better in homeschool? It's very difficult to compare with a hypothetical.
Also, many of the positive outcomes are self-selecting. For example, homeschoolers who don't want to go to college don't often bother with standardized tests. Those who aren't good spellers don't try to enter spelling bees.
As far as professional training goes, I have read many anecdotal accounts of public school teachers who found their training almost useless when they decided to homeschool their own children. As Fezzik from Princess Bride said, "You use different moves when you're fighting half a dozen people, than when you only have to be worried about one." Contrary to popular belief, most homeschoolers don't have dozens of children :-)
For just one example, grades are utterly irrelevant in homeschool. You either keep working with the child until he achieves mastery, or you drop the topic and come back to it later when he's better prepared.
When we started homeschooling, I took it upon myself to learn as much as I could about education: how children learn best, how to deal with ADHD, etc. I learn from the same sources a professional educator would. To my great surprise, I found that schools often can't or won't follow their own experts' advice.
When I asked professional educators I know, I found this was an extreme source of frustration to them. They often know methods that would help their students learn better, but their hands are often tied in being able to employ them, because of political or practical considerations. Ironically, homeschoolers are in a much better position to utilize those expert recommendations than professionals are.
For example, I discovered that experts know a 10-minute recess once per hour would solve a good part of the behavior problems ADHD causes in schools. My son's school had one 15-minute recess per day, which he was often held in from due to his ADHD-related behavior problems. Now that he's homeschooled, it's almost a non-issue. He can take as long of a recess as he needs, whenever he needs it.
Some other examples where we're following the professionals' advice better than the professionals can:
- Each topic is geared to the individual level of challenge they need, at the individual pace they need.
- We can go into depth on topics our kids are passionate and curious about at the moment. My seven year-old probably knows more about ancient Egypt than I do.
- We use a computer program that gives instant feedback for math practice, where he often used to fill out entire worksheets incorrectly and not know until the next day or even the next week.
- Our differentiated instruction is able to be much more targeted.
- Our kids have many more opportunities to ask clarifying questions. We never have to move on in order to keep to the schedule or to give other kids a chance.
- Our kids are not overloaded with "homework." They have plenty of time for play, the arts, and family activities.
Do some homeschoolers have worse outcomes than they would have had at public school? Sure, but the statistics are in homeschooling's favor, and I think the outlier cases are mostly due to ideology rather than a lack of training.