In public you would carry a wet bag to put your dirty diapers in. You could leave the poop in and just deal with it at home, or if it's solid, just dump it in the toilet. There are also flushable liners that you can buy, so you are just flushing the poop.
There are different types, from prefolds or flats with covers to all in ones that work just like a disposable diaper. For exclusively breastfed babies, you just throw it in the wash, breast milk poop is water soluble. Once your little one starts solids, you have to flush the poop ( your supposed to do this with disposable diapers too). A diaper sprayer or spatula helps if it's not solid.