I've seen plastic used under the sand, but I wouldn't recommend it as it tends to make the walkway less stable and it can trap moisture underneath the walkway and cause freezing expansion issues. A properly constructed brick or paver walkway will shed most of the water that hits it without additional waterproofing. Where they've sunk over the ages you will often see puddles forming on them. Some water will soak into the sand, but that will happen anyway if the plastic is under the sand. They actually make special "permeable" pavers that are designed to let water through for areas where drainage is a problem.
If I really wanted the pavement to be completely waterproof, I'd apply some sort of sealant after it was built, but of
course, that usually involves plastic-like chemicals. This is most often done not so much to make the pavement waterproof as to protect it from stains caused by fluids leaking from vehicles.