RM,
I too am working on potty training, with my son, and use a non-flushable wipe. However, he is still wearing pull-ups or diapers so after we go pee on the potty and dump the poop in the big toilet I wrap the wipes with the diaper. Then drop them in the plastic bags the newspaper comes in or the plastic bags I use for fruits & veggies from the supermarket. After tying a knot, folding the bag over itself and retying I dump into a garbage can in the bathroom. Eliminates most odors and doesn't ruin plumbing.
Good luck with your adventure!
PS I buy the Walmart brand to keep for the adults or occasional use for my son and they're really too thin to do a good job on his bottom. I personally prefer using baby wipes to ensure I get him clean.