First things first, poop is something that needs to be composted outside. Indoor composting methods (to be covered soon, I promise, especially with winter coming) aren't recommended by this writer, at least not until more experimentation is completed in a successful manner. A lot of dog owners have a yard of some sort, but if this isn't the case for you, try engaging in a little bit of guerrilla composting by building your pet poop composter in an abandoned lot, earthy alleyway, under a camouflaging shrub, or in a friendly neighbor's yard. Only do that last one if your neighbor knows about it and agrees, otherwise you could end up explaining to a judge why composting trumps trespassing; it's true that the same situation could arise from any of the other spots, but those are usually less monitored, and so easier to get away with.
It's easiest to build a pet poop composter with an old trash can or some chicken wire. This isn't necessary, but it helps keep the pit from collapsing in on itself, as well as makes the finished compost easier to haul out. Cut the bottom out of the trash can (tin snips, also called aviator snips, work great for this), and drill or pound a lot of holes in the sides to promote drainage and interaction with the surrounding soil. Dig a hole a little deeper than your trash can or chicken wire. If you have heavy clay soil you'll want the hole to be a foot or more deep than your lining material, and then add some gravel to help with drainage. To tell if you have clay soil grab a handful of it when it's damp, roll it into a snake, and bend it into a 'U' shape. If you can do this without it breaking, you have clay soil. This method is stolen from the amazing Alys Fowler, who wrote a wonderful book on gardening and composting which you should definitely check out here.
If you're working with chicken wire, line the hole with it, keeping the wire level with the ground. For a trash can, sink it into the hole, but leave two inches sticking up above ground level to accommodate the lid. Now just collect all the animal waste in your yard/litter box/terrarium/cage/aquarium and dump it in. Some people like to throw a little septic starter in to promote healthy bacterial growth, but I just toss in a little topsoil and call it good.
I hadn't done much more than this in previous years, but after reading the humanure handbook I now cover the poopy up with leaves or grass clippings. This keeps the smell down, but also contributes much-needed carbon material to the nitrogen-rich animal leavings. It make for better compost, eliminates the extra step of mixing pet compost with regular compost for balance, and allows me to open the composter on the hottest of days with the freshest of turds and not have to worry about the stink. In the summer I like to water the bin, but I live in a desert, so this may not be necessary for those residing in moister climates.
If you used a trash can, just the leave the lid on when you're not adding material to the composter to keep people and animals from falling in. You can also cover the hole with some scrap lumber, a bit of old carpet, or leave it open, whichever appeals to your aesthetic sensibilities. I do recommend covering it or cordoning it off in some way, as my cute but clumsy shih tzu-poodle fell in once and made quite a mess afterwards while trying to get out. You may not have the same dog, but there are clumsy things everywhere who don't deserve to fall in what is essentially a giant toilet.
Poop does carry pathogens, so it's important to let the compost sit, undisturbed, for a year or more in order to make sure every thing dies off. I do this by building two composters, then filling one up while the other one sits for a while. If you are thermophilicly composting your pet poop, this waiting period is not necessary. You can take extra precaution by only using the resulting compost on horticultural gardens, rather than agricultural ones, but I've never had a problem as long as I've either let the compost sit for a year or put it through a thermophilic process.
Finally, it's important to put only biodegradable material in your pet poop composter. Of course any feces or urine is fine to go in, but make sure the collection method is also compostable. I use Biobags to collect dog droppings when we're on a walk, and have found that they compost just fine. If you have a cat, opt for cat litter made out of corn cobs, newspaper, saw dust, or wheat husks, as all of these materials work great in compost piles; sand, clay, and crystalline based kitty litters are a no-no as they can leach toxins and mess up your soil structure. Fish tank water is fine to pitch straight onto your garden (roses love it), but can also be poured into the pet poop composter. If you have hermit crabs or other small animals try lining their tank either with sawdust or coconut husks, as these are earth-friendly beddings that compost with ease.
As you can see, it will take an initial investment of effort build a pet poop composter, and requires developing some new habits when dealing with your pet's leavings, but once these routines are established I think you'll find dealing with pet waste a much easier, pleasanter, and rewarding task than it ever was before.