I used to burn up most of my junk mail and such in my wood stove in the winter. However things like stacks of papers and catalogs don't really burn well, you have to keep stirring them up to separate the pages so they will burn. I also used crumpled newspapers to start fires. The downside was that crumpling them leaves your hands all black from the ink. Eventually I came up with the idea of shredding the stuff and then using it to start fires since it burns way better when cut up. So I saved bags of shredded paper for starting fires in the winter. Problem was that I was shredding way more than I was burning. So several years ago in the spring I decided to try tilling the stuff into the garden. At one time this was a no no because the inks were lead based and there were other not so good things in the paper. But since recycling paper took off, the inks have been cleaned up and many are soy based and so are harmless.
The shredded paper just disappears in a short time. The little bit left on the surface even degrades. The paper helps hold moisture in the ground and when it decays it adds nutrients to the soil. It breaks down way faster in your garden than it does in landfills and you get the benefits of the organic matter. In addition I have noticed that my earthworm population has increased a lot. I guess they are working at being educated with all that junk mail. And for those of you who are concerned about things like identity theft, nothing leaves your property, its completely gone, turned into compost.