Pros: Recyclable, resists water, handles are built in, can be made of recycled material.
Cons: New plastic is made of oil (which is toxic); plastic takes decades (even centuries) to decay; plastic is harmful to animals (even decaying plastic); the oil it's made from is not renewable.
Cons: Usually have no handles, and when they do the handles suck; made of trees; gets weak when wet; is made from living beings.
Pros: Recyclable; trees its made from are usually grown specifically for paper, with replacements planted soon after; it's renewable; can be made of recycled material; is a renewable resource; can be made of things other than trees, such as hemp;
Either option is valid. The only remaining considerations are weather and transportation. If you have a car, or something to carry paper bagged groceries in, you're okay. If you have to walk any distance at all (even to the bus), then the crappy handles paper bags have point toward "plastic." Especially here in Portland, where it rains a lot for half the year; I gave up on paper bags ages ago for these two reasons. Paper bags aren't a good idea when you have to walk 30 blocks in the rain.
The best option remains reusable bags. I use these when I go out with the intention of grocery shopping. If I'm just out and about and can fit the stuff I get on a whim into my big blue bag that I carry everywhere, I do. If not, I use plastic. I even have one reusable bag that is entirely cloth, and can be rolled up, so I could take that everywhere with me if I wanted to. And my purse handle is broken, so I've been carrying that in a cloth library bag.