Earlier quoted context omitted.
Stores should provide non plastic bags and give you a large discount for returning/reusing them.
The trouble with paper bags is that they don't biodegrade without oxygen, which is what happens when you bury paper under tons of other material in a land fill. So the paper and the plastic are both there 100 years later, but the paper takes up more space in the long run because it doesn't compress as small. Unfortunately there's no good answer here yet that I've seen.
In italy those have been the norm for many years. They actually seem to decay quite fast (i.e. you can't reuse them for too many times).