Hi niels_olson,
I wasn't homeschooled at all. I went to private school through high-school (around Charlottesville, VA, in case you're in that area) and then did public university. I went to school with some home schooled students. Some were good, some had issues. Most of them had at least a few social issues, but then, I think any way you sliced my high-school population, the majority of a given section had some issues; such is the nature of being in high-school.
The point is, this doesn't really tell you anything useful. I could tell you what my gut tells me about homeschooling, but that's not going to be especially useful either. Instead, I'll share two things that I think might actually be of some use to you on this issue.
One: you're asking a community of amateurs. The people on this website are, at least taken in aggregate, a standard deviation or two above the norm at what they're good at. However, I have met very few people who're particularly smart at everything. Instead, people are usually smart about a small number of things. The most wise people are usually wise enough to know they don't know squat about the rest. Therefor, I would suggest that there will be lots of well-meaning, reasonably smart people here, I'm not sure that the people who actually know what they're talking about are going to come out over the people who don't know enough to know they don't know what they're talking about.
And what makes it worse is that studies show, if you don't know much about this, you're going to have a hard time telling the know-it-alls from the know-nothings.
Two: you're asking a self-selected group about their experiences, and as we all know, the plural of anecdote is not data. Again, a lot of people on here are smart, and on top of that, highly (self-)motivated, people. These are good qualities that I'm sure you want to give your children, but you're probably likely to get more success stories about anything with a reasonable success rate here, given the population you ask. If I ask a room full of millionaires how many people eat raisin bran and I get a number of positive responses, I can't then conclude that iron and dietary fiber have a greater-than-average incidence of financial prosperity.
I'm not suggesting that it's not worthwhile to hear what people have to say about their experiences; I'm just urging you to take what you read with a grain of salt and to seek out data about homeschooling. Try to find non-biased studies on it, compare the costs to comparable options (Montessori, local alternative private schools), and then ultimately apply your best judgment.
Good luck!