Here's Rachel's hack, which happens to describe exactly how I've used those newspaper bags for the last couple of years:
If your newspaper comes in a plastic bag to keep it dry, don’t throw it away. We save ours and use them to wrap up soiled clothes on the go, dirty diapers at home, and other kid messiness! For five years we’ve wrapped every dirty diaper in these bags; put it in, twist it tight, double the bag over again and tie it shut and it works as well – and cheaper – than a Diaper Genie. New parents always think that they need to buy special diaper containing gizmos – but you really don’t!
Andi also suggested Press'n Seal as another diaper disposal alternative. Too expensive to use every day, but good in a pinch.