I've seen that type of behavior also in some kids. Not just homeschooled kids, but also kids in the very gifted track.
My family was at a large birthday party yesterday and there was a girl about 12 or so (I have NO idea how she was schooled). There was something very disconcerting about how she would insert herself into adult conversations with an arrogant tone of how she could solve any given problem. At 12 or so, one does not have the life experiences to even offer an opinion on many subjects, yet she thought her opinion was wanted, needed and gave it many times. She actually shouted at my husband at one point who had been into a conversation with another adult and mentioned how the founder of Greenpeace recently endorsed the use of nuclear power in a WaPo article. She told my husband - a 48 year old man - that he was an idiot for stating this and then stomped off telling someone else how stupid people are. Supreme arrogance.
This is not the first time I've seen this type of behavior among young and middle teens and the only connection I have found is that they are either homeschooled or in the gifted program at school. So these kids may be smart beyond their years, but they are dumb as dirt in terms of respect and the ability to admit they might just be wrong.
My experience has been that generally kids are arrogant when they aren't taught humility, no matter how they're schooled. But I have yet to come across a homeschooled kid as arrogant and rude as the girl you described. (Based on the description you gave, I wouldn't assume that she's either homeschooled or gifted).
My oldest child really is very bright in terms of his amount of knowledge on certain topics. He really does know much more than the average adult (including his mother here) on a number of topics. But I always hammered humility into him - in fact, I went overboard. Most other kids are far more arrogant without knowing as much.
However, I have no trouble at all with him having more knowledge than I do. I'm happy about it. In fact, if I'm not sure about something, I ask him about it, and he's only 10.
You hit a chord with me. I don't know if you are familiar with Dennis Prager or not. He is a radio talkshow host and I happen to love him. He speaks frequently about this phenomenon. Not that the kids are necessarily bright, just that they seem to think they can just go up to adults and criticize them or talk to them as if they are "peers." He always makes such excellent points. The real problem is parents who allow their children to comment to adults as if they were also adults. Some think it's cute. I've seen it. They have not earned the right or status yet, but they are often not scolded or disciplined for this behavior.