It should be fine for traffic after 6 hours or so. For horse and dog walking I would wait until the next day. Not so much for the horse/dog safety but for the surface of the road. I assume you care, since it is your road.
"We can judge the heart of a man by his treatment of animals." ~Immanuel Kant
You can walk on it while it's still hot from the paver, but I wouldn't advise doing it in rubber boots or subjecting the dog's feet to that. Give it the six hours. Our road got paved and we experimented a little, so my DH's waffle boot print is forever memorialized at the side of the road. The edges aren't usually compacted totally, and if they make those preformed curbs they are very soft and will take prints for a while, like a day.