Well, yes, fear is contagious, as anyone will tell you who has been stuck in a lift with a claustrophobic. But I'm not sure that it crosses the species line. Dogs chase a frightened person because chasing is fun. It may have nothing to do with fear being contagious or even having a scent of its own. But while we're at it, I think fear does have a smell of its own.
Many people say they can smell fear in other people.
It makes sense. Fear produces a rush of adrenalin, and an overload of any chemical in the blood will probably affect the composition of substances secreted, like sweat. So they know when we're afraid. It depends on the nature of the animal concerned . The dog might attack, the horse would probably prefer to bolt.