For the SATs, homeschooled students score ~75 points higher than public school students. We can draw only limited conclusions from that, though. People self select into taking the SATs. And there's the the bigger issue of demographics: given the demographics of homeschoolers who take the SAT, you'd expect them to outperform anyway. Regardless, more data is better.
I would say that "in some states" is the key here. A lot can change depending on the state.
Which has essentially the same incentives and results that the NRA does -- cater to the craziest of your constituents, fan outrage, then push the most extreme views (to protect everyone).
As a result, HSLDA essentially says that no child abuse or sexual assault exists in the home schooling communities, because it's the only way to justify their position that zero regulation is the only appropriate amount of regulation.
Reality, of course, has a non-zero rate of abuse (which the HSLDA then does their best to hide). Warning, pretty terrible stuff: https://www.patheos.com/blogs/lovejoyfeminism/2013/05/hslda-...
This makes the comparison very hard, since one side is has all of its successes and failures critiqued in public, and the other side is only ever talked about with anecdotal data.
Personally I believe there should be some sort of periodic standardized test for homeschooled kids too. Even if it's not for "passing" a grade, so we know in which level of learning they are compared to everyone else.
If we can't compare them, how can we evaluate its merits and flaws?