I put my meat in the thin plastic bags and my veggies in the thin plastic bags - then everything goes in my reusable S - which I wash out once a week.
Yes, the store keeps the money but what they really want to do is encourage people to use reusables instead. If I forget to bring a bag with me to the store I always ask for a paper one. At least those are biodegradable and can be go into my recycling bin with all the other paper.