Not all curbside recycling programs take shredded paper, particularly those that have single-stream recycling. The shreds can clog up the machinery that separates the recyclables. If you can’t recycle your shredded paper, what can you do with it besides put it in the trash to give it at least one more use?
Use it as packing material. It can cushion breakable, valuable items when you’re storing them or when you’re moving. Put it in under kitty litter. Since cats like depth, it can add some depth to the litter box, meaning you’ll need to use less litter.
Loosely stuff it into empty toilet paper rolls or cut up paper towel rolls to make kindling or fire starters. Make more paper. Stuff it in an opaque vase to keep silk flower arrangements in place.
Donate it to an animal shelter where they can use it for bedding or litter, but call first to make sure they need it. Use as filler for gift bags.
Use it as a layer of mulch to underfill dirt, compost, or other mulch. This works best with plain white paper or newspaper. Colored paper or paper that’s glossy won’t break down as easily.
Make paper clay. This video below demonstrates how to make DIY shredded paper clay that can be hand molded or used in molds.
If you don’t want to do anything with it, offer it on Freecycle or a Facebook neighborhood swap site. There may be others who may want it for any of the above reasons.