Most people don't bother to do anything special with organic waste. It goes with the rest of the trash.
I use them in my house for organic waste. We actually divert about a green bin full (the tall ones they give houses) of our trash into the green bin program because we can easily put it all in the plastic bags we get some of the time when we go shopping. (we also use reusable bags when we shop). It seems to me like a behavioral consequence of a ban would've been for people to not participate in the green bin program fully.
Ideally people would be willing to pay for bags but I don't see it happening. I would like to see some kind of mandate though that would have the plastic bags being used being altered in some way so that they don't clog up the recycling machines, and are more biodegradable than they currently are.
Edit: Too elaborate a little, what happens is people put the plastic bag inside the bins in their house where they throw their organic waste. By doing this there is no residue on the bins themselves that end up rotting, smelling, and becoming a health concern.