America's situation is different; they have an idealogical opposition to social welfare and there is no system to opt out of. The homeless situation in SF is far worse than in any EU city I know of.
I don't know if I'd really call it a situation. Homeless people choose to go to SF because it's one of the best places for them. You won't freeze sleeping on the street, plenty of handouts...
Anyway, whatever the reason, I think it is actually a "situation". Many cities have homeless problems, but SF's is amazingly in your face. Most SF people I know would simply not walk through Tenderloin at any time, even in broad daylight. It's in the middle of the freaking city! That is not normal.
I've only been in the TL for three years or so, but I've spent the majority of that time wondering how regular working-class families with kids can afford to live in the area. I can't imagine how street-level pushers or prostitutes make rent. Maybe they rent places out in Hunters Point or Outer Sunset?
http://www.psychologytoday.com/blog/the-everyman-psychiatris...
I recently lived in the Tenderloin for six months, and my impression was the the majority of the chronically homeless there are crazy.
Union square has a lot more alcoholics and drug users, and 'talented' homeless people like street performers and pick-pockets. They may have some minor issues but they're clearly there by choice.
Like Copenhagen I believe it is extremely intermingled with substance addiction problems, especially alcohol. It's a hard nut to crack, that's for sure.
If you are a legal resident you will be offered a place to live, as well as money to survive, if you have the wherewithal to go to the social services. The ones who don't use this are usually mentally ill or too deep into substance abuse of one sort or another.
The thousands you are talking about are probably illegal aliens, who are obviously not offered the same services that residents enjoy. Can't really blame Copenhagen that thousands of people without resources or prospects stream into the city from other countries. In either case, they all reside in government or privately run shelters with beds and meals available, which is I guess is why we don't see them in the street the way you do in New York and San Francisco.