And, yes. That would make sense so that the poor dogs didn't get trampled by the horses! And the dog and horse owners can drink out of a third side in safety.
Yes, it does make sense that dogs, horses and people would drink from three different sides so they don't bump into each other. And it just occurred to me (slapping muh forehead, DOH!) that, in addition to a faucet, people would also have a little cast iron bowl so the water didn't spill onto their feet, just like any other water fountain.