We are hoarders of paper at our house. Scrap artwork? Uh, yeah. Envelope? Yes, please. Cereal box? Oh yes! Cardboard box? Yay!!!
I pretty much never get unused paper, and even that is secondhand or promotional notepads or discards. The kids are used to drawing on their own old artwork, and go to the paper bin for anything from magazines to old school worksheets. They re-color old coloring, turn it into origami, then paper airplanes, then for collages. Kids can understand that paper comes from trees, so it's good to use less and help them live - for shade, birds, fruit, and more.
I, in turn, save those leftovers for notepads or grocery lists or journal paper. When they finally make it to the recycle bin, they are sometimes rescued for making new paper, projects, origami - or they make their way out to be turned into new paper.