Both are recyclable, although check with your city or village streets & san to see if they allow the plastic variety in with other recycling. If not, your grocery store will have a drop off box for recycling.
The environmental footprint of both is a toss up: chop down more trees for the production of paper or pollute the air and use more oil for the production of plastic. Maybe it comes down to personal preference--I prefer to save the rainforests. At the same time, I'm really trying to cut down on our family's use of other plastic items to help with the air.
I also prefer the plastic because we have a rather large dog, and we use a few on walks. Back when the kids were in diapers, we also used those plastic grocery bags for the disposal of rather disgusting items (tied the bags shut and took out the to trash right away!).
Paper is great for loading with newspapers, junk mail and soda cans for recycling.