Ever tried Bokashi? It is the best organic breakdown method I know of for meat and carnivore wastes or when in sub-ideal composting conditions (i.e. too wet or dry or acidic etc). This is because it is produced by a the fermenting of a robust, diverse ecosystem (ideally the most diverse
compost tea you can make). In doing so you are selecting out the most badass decomposers in the bunch and are multiplying them by combining the tea with grain and molasses in an anaerobic environment. You do not need to buy the EM or other stuff sold as bokashi starter if you can make a good aerobic compost tea. I use duck pond water, molasses and wormcastings aerated for 24-48hrs as a good base tea. I then add more molasses to warm water (I use recipes online just make my own microbial colony) to continue the replication of organisms, mix with the tea at about equal parts, and mix with wheat bran (you can use rice or barley or any grain with good protein content but this is cheapest here) to a dry oatmeal consistency (just a drop or two of moisture should come out when squeezed firmly). This is then firmly packed in a sealed bucket for 2-4 weeks (shorter is fine but longer breeds even tougher microbes, too a point). It should smell like wheat beer when done, sweet and fermented, with a white network of fungus. Blackish or green fungus is a bad sign and I toss it under my established evergreens. When done the bokashi can then be air dried and stored indefinitely. I spread about a 50lb bag's worth over my 1/2acre every spring after the heaviest rains have passed. This gets everything breaking down very quickly, whether leaves or straw mulch or dog and bird poo. I also use it to inoculate compost teas, birds' water, mulch and soil with some great, tough bacteria and fungus that is beneficial for pretty much everything people like. It will not however, turn a bucket of dog poo into anything nice. I tried :p
I would also try to bury dog and cat poo well beneath 4-6" of organic matter or dirt, ideally where non-food bearing trees will absorb any runoff before it hits a stream or drain.