I'm not sure how it's done in Australia, but here in Canada the Queen rubberstamps the appointment as Governor General of whomever the prime minister and cabinet recommend, and this person exercises "all the powers, authorities and functions of the Queen in right of Canada". The G/G has been a Canadian citizen since 1952 (previously it was usually a British noble, diplomat or ex-military man), and except for a few questionable appointments in the 1980s and 1990s, has been apolitical and generally well thought-of.
Though there have been calls for an affirming vote by the Senate and House of Commons, this has never been done.