The process for digesting toilet paper in most water sewage systems is synonymous to the way your poop is digested. Hear me out.
When you think about your poop you're never like "man, I'm throwing away my poop, what a waste!" You assume that your city has a reliable septic system that will dispose of your human waste without any lingering smells, diseases and sightings. And it does!
Well, it's the same process with toilet paper, and that's why as a zero waster, I can rest easy using toilet paper.
Just like human waste, toilet paper is dissolved, gassified and filter through a three-step process to send clean water back into waterways.
Toilet paper is patented to dissolve, because the fibers are small, making it one of the most fragile pieces of paper.
Toilet paper is made with short fibers, compared to facial tissues or writing paper. The short fibers make the paper softer, and so it dissolves quickly in water.
By the way, this is why you DEFINITELY SHOULD NOT BE FLUSHING BABY WIPES, which are known for their sturdiness and made of interlocked fibers. Cities, like my hometown Atlanta, have launched campaigns-- #nowipesinpipes- to remind residents to NOT flush their wipes.
Also, don't flush anything that isn't human waste or toilet paper. No tampons, sanitary napkins, diapers, condoms, etc.
Atlanta's centralized municipal waste treatment uses aeration tanks which "optimizes biological carbon oxidation, nitrification and phosphorus removal, the addition of secondary clarifiers, effluent filtration, ultraviolet disinfection and post-aeration" to breakdown waste (and toilet paper!). Source
This is a three tier treatment system, the first sorts out the large debris, including tree branches and leaves, floating trash, dead animals, and clumps of paper, and such.
The second tier is the aeration chamber, which fills the chamber with bacteria-filled air to breakdown waste.
The last tier filters and treats water before sending it down the Chattahoochie River.
As for the cardboard rolls and plastic bags that toilet paper comes in, I recycle, which isn't optimal for a zero waster, but suitable.