The carbon foot print is debatable, but plastic in the environment is basically there forever, where a paper bag breaks down relatively quickly and natural processes can deal with cellulose.

I'd moved into a house and noticed a plastic bag stuck on a branch high up on a tree. When I moved 11 years later, the bag was still there, showing essentially no signs of deterioration, even after 11+ years of exposure to sunlight and seasons that vary from -35C to +35C.

Paper bags can be used as garbage bags as long as the garbage isn't soup. Mum did it back in the 1960s and 1970s until plastic bags replaced the traditional paper bag. We're both back to using paper bags for garbage.

The last plastic bags that my grocery store used were so thin that they almost always had holes in them and leaked, so there weren't appreciably better in that respect than paper.