In downtown Shanghai I got to watch a mother walk out of her business holding her child, have the child crap on the sidewalk (while being held in the mothers' arms), and then brought back inside. This was not a baby. The kid looked about 3-4 years old.
This goes on in subway stations as well. The parent has the kid sit on the edge of a garbage can and crap into it.
If you go you'll see the children commonly wear these lower garments with the whole groin area open. It's so the child can squat and pee or poo anywhere without needing adult attention. The child is taken to public parts in the city and around downtown like this.
Chinese public conscientiousness is far below anything you can imagine until you experience it.
A strong agrarian tradition where no piece of human waste is going to be left festering in the sun - somebody is going to pick it up and make good use of it. In all seriousness this has been the case until chemical fertilisers became cheap enough, before then every town had organised dung-gatherer guilds who organised daily cleanup of faeces in the street and sold the product to the farmers in rural areas. Even urine-soaked earth dug up in alleys can be sold as a source of nitrates.
A second and more ingrained reason is that young children are regarded to only have the intelligence of animals because you cannot reason with them like how pre-Christianity Romans thought, thus they are to be treated with the same level of respect that animals deserve: if kids want to relieve themselves, then it might as well happen on the spot because self-control is beyond their facilities.
Shanghai is also unusual among major Chinese cities that many households in the old town are not serviced by sewage lines. Instead everybody bought their nightly waste to public septic tanks and pour it down a chute before a vacumm truck takes the load away every few weeks. Thus the residents are typically less fazed at the sight of faecal matter than their compatriots.
I'd say it's more to do with development than anything else.
Having been to Taipei several times myself, I'd say that it is no better or worse than any Asian city. but that's really just prejudice. I will concede, however, that Taiwan had the value of hygiene worked into them during the Japanese rule. In the first ten years of the 20th century, there was 49 newspaper reports of public urination leading to a formal warning by the police in Taihoku alone (ironically, more than half of these high-profile cases were perpetuated by Japanese), and public record shows that thousands of people of all sorts of background were fined every year for the same offense that did not make it to the news.[0] This really goes to show that the colonial government took the concept of hygiene very seriously, but even today some people will just do as they please.[1]
[0]: https://books.google.com/books?id=AKi5Wm5Mu_UC, page 27
[1]: http://news.cts.com.tw/cts/general/201001/201001030380454.ht...
Do you have any other info on the second point?
http://dc.cod.edu/cgi/viewcontent.cgi?article=1007&context=n...
In short, Confucian scholars view personhood not as something gifted at conception or birth, but a learned virtue that develops progressively as each person lives through different stages of their lives. Thus rearing young children is not unlike taming a wild animal and a lot of nasty things kids do can be forgiven on the basis that they have not learned the proper way to act yet.
The ugly side of this thinking is that this naturally restricts youth rights since they are saw as an extension of the will of their parents; indeed one can argue that children are never fully independent until the death of their parents absolves them of these control and responsibilities. On the other hand, Confucianism has always been ambivalent if not indifferent to topics very touchy in the Judeo-Christian tradition such as abortion and infanticide, because the victims have not had a chance to develop their personhood yet, and thus these acts are sometimes considered deplorable but never compared to murder.
http://www.vancitybuzz.com/2013/08/meanwhile-in-richmond/
http://www.theglobeandmail.com/life/the-hot-button/the-boy-t...
http://www.cbc.ca/news/canada/british-columbia/boy-peeing-in...
http://www.straight.com/news/417596/culture-clash-over-kid-p...
It seems to be common enough: https://www.quora.com/Chinese-Etiquette-and-Behavior-Is-poop...
Better than doing it in one's own drinking water, but I think what saves the Chinese is not education, but a tradition of cooking absolutely everything (see sibling posts).