The most common material used by pet owners to clean up after their pets is reused plastic packaging, grocery or bread bags, newspaper sleeves, and other similar bags - and those are all plastic!
One could choose more "eco-friendly" bags that are compostable or biodegradable to use instead but, even though their production may be better (which is a great plus!), they will still be going to the landfill the same way all the other trash goes and, since landfills are designed NOT to have much decomposition happen to anything that is buried in there, the bags will not disappear; they will be spending their time buried in the landfill together with all the other trash.
Other alternatives are products made of paper (like recycled paper and bamboo bags or ScoopEasy), or reusable dog waste bag (like Poof), and also flushable bags* (like FlushPuppies).
Choosing one of these products over regular plastic depends on your preference, routine, and how much money you would like to spend on such solution.
I wonder how this job was done before plastic bags came along. That would be an interesting research to be done for a future article.
*IMPORTANT information obtained from Brevard County Utility Services Department and South Central Treatment Plant (http://www.brevardfl.gov/UtilityServices/ContactInfo): In Brevard County, you can flush pet waste but flushable bags are not 100% guaranteed to work - it depends on the composition of the bags and their requirements for biodegrading. So FLUSH THE WASTE but NOT THE BAG.
Sources:
http://www.onegreenplanet.org/environment/eco-friendly-way-to-dispose-dog-poop/
http://www.dogster.com/lifestyle/5-alternatives-to-a-traditional-poop-bag-including-a-reusable-one
https://thedailyshep.com/how-to-pick-up-dog-poop-without-plastic-bags/
https://thedailyshep.com/how-to-pick-up-dog-poop-without-plastic-bags/