We don't really have a choice here of "paper or plastic," as it's almost completely plastic here. So, I use the plastic ones in some of the smaller wastebaskets in the house. The rest go into a separate one to be taken back to the recycle bin at the market, along with all the other plastic bags (bulk produce, bread wrappers, celery/carrot bags, etc.). For nearly 30 years I've been using canvas and rip-stop nylon bags when I shop. As Vera does, they go right back into the car as soon as everything's brought into the house.
Since we recycle so much, I only empty the kitchen trash can (13 gallon) about once every three weeks. So, we only use a few plastic trash bags per year this way. It's been amazing how little goes out since we've been seriously recycling.
All our newspapers, paper bags, cardboard boxes (cereal, cracker, etc.) are torn up and used as "kindling" in our wood stove during the cold weather months.
Everything from the paper shredder goes to dry matter in the compost bin.
Oops, we were only talking about plastic or paper bags. My bad. I'll go now.