The main concept here is to have a standard training curriculum for all animals that are going to be allowed at public places. This will include teaching the animals where not to defecate, make noise, or show unnecessary aggression, etc.
One can train their pets using reinforcement methods that involve reward and punishment. Reward is a positive form of feedback that encourages a desired pattern while punishment is negative feedback that discourages an unwanted behavior or action. For example, you can give a dog a food treat when it poops at your preferred spot (reward) or you can yell at the dog when it poops on your door mat (assuming that is unpleasant to everyone). Yelling at the dog is a form of negative feed back.
Animals that have been trained to recognize standard symbols will be able to tell where not to defecate, make noise or do any other 'unpleasant' things in some places just like humans can tell where not to make calls or drive fast when they see the relevant signs no matter their language.
The training system can be extended, hopefully, even to wild animals so that animals conform more with human societies. People will definitely leave more room for animals if they do not dirty our favorite places,damage precious objects, or interrupt our activities. Moreso, if they were to become useful in a society, the rate of consumption of animals as food is likely to reduce.