I haven't read through all the responses (sorry) so just thought I'd add what I do ...
I have a garbage can that I throw very small scraps into that I know I won't use or share with anyone else. I line the garbage can with an old pillow case (reminder to go to Goodwill and get more pillowcases). When I have enough scraps, I sew up the end of the pillow case and DH takes it to a local pet shelter.
I have larger scraps that are becoming a problem though ...