In my language we have separate words to chase off each (common) animal, like "tchibe" for dogs, "sheetz" for cats, and "ish" for birds. I can think of only one corresponding word in English: one time I saw Marge from "The Simpsons" telling her cat to "shoo". Is that the word I'm looking for? Could you use it as a mildly insulting slang (more fun than insulting) to address another person?
Shoo works for animals and people. It is, as you suggested, mildly insulting to tell a person to shoo. When telling a person to shoo, there is usually another form of telling them to go away accompanied with it:
Shoo! Go away!
If using it in a more playful manner, you could say:
Shoo! Go on, shoo!
Which is accompanied by the "go away" hand gesture (the opposite of "come here"). The hand gesture is not required.
Alternatives that could be used:
- Go away
- Leave me alone
- Be gone (uncommon)
Shoo is adequate, but you could also use:
Go away!: used to drive away unwanted or annoying people, animals, etc.
And probably it could be applied for people (e.g. asking someone unwanted to leave).