Many malls and major department stores already have recycling areas that are accessible to customers. The store I work at has one specifically for plastic bags right near the front that takes bags from any store and not just their own. Sure, It may take the plastic that is in bags 10-20 yrs to biodegrade but that just means it's material can be reused for that long. If we compare that with paper bags, It takes far less time for them to degrade so you end up requiring more wood pulp to manufacture them.
Plastic bags are non-biodegradable and hence it will take forever for it to disintegrate. New landfill areas will be a necessity as time passes as the plastic bags will accumulate in landfills. Another alternative would be to practice open burning of plastic wastes but this will lead to air pollution. The gases that are produced is toxic and could lead to respiratory diseases such as asthma or pneumonia.