Shitting in the streets/public places, definitely NOT COOL!
But with my kid we did something called EC where he went diaperless while we were at home or in a "safe" place. We got really good at catching the warning signs and hustling him over to the potty when he had to go. Only had a couple of mishaps. He was potty trained before he turned 2.
If we were in public, he always had a diaper on. We would still try to get him to the potty if we were in a restaurant or something, but if we didn't make it or if there was no potty nearby, we still had the diaper for backup.