Your basic premise is wrong. The police are allowed to arrest such people. No, you have no vigilante rights to go shooting people who are committing crimes to others.
To get some idea of how people feel, start here.
For opinions from police officers, start here.
For a possible solution, start here.
I live in the Seattle area, and what the OP refers to has been in our local news A LOT. Every time I go into Seattle, I see the homeless camps and they are filthy.
Retired Cal Cop Sergeant & Teacher
Walk humbly with your God
-- Courageous, by Casting Crowns ... http://www.youtube.com/watch?v=pkM-gDcmJeM