As mentioned in For people, can you say "a British" like you can say "an Australian"?, you can use "an Australian" to talk about an Australian person.
But is it also ok to use "an Australian person"? If not, is it merely too verbose, or does adding "person" make the language seem more cold and impersonal?
What about "an Aussie person"? Would that be even more unidiomatic, because you've got a word that's a shortening and is slang before a redundant word?
Also, is "an Australian person" being unidiomatic likely to change in the near future? How did "Jew" become pejorative? talks about how using a noun instead of an adjective is sometimes regarded as offensive nowadays.
I tried looking up "Australian person" at onelook.com, but didn't get any hits. That in itself doesn't indicate anything, as onelooking "British person" only got an Urban Dictionary result.