Local classrooms: Elementary, middle, and high schools often have classroom activities using newspapers: current events, collages, language/story-writing skills, and many others. Public libraries can also benefit from having extra copies of current papers.
Around the house: Whether it's papier-mache with the kids, cleaning windows, a drop cloth for messy hands-on meals (crab legs or BBQ), painting projects, or garden mulch, look for reusable options at home.
Animal shelters: Absorbent newspaper (the majority of the paper–glossy coupon inserts aren't very effective) can be great for kennel liners. Call your local animal shelter and see if they can use your weekly extras.
Neighbors: Have friend, family, or neighbors who love to read the paper but aren't into coupons? Take out your inserts and let them have the rest after you're finished.
Recycle: Lessen your environmental impact by recycling newspaper instead of throwing away. Call your city/county to learn options for recycling programs or drop-off locations.