When a Commonwealth country such as Canada or Australia has a Governor-General, is he or she expected to use his/her own judgment with respect to issues that come before him/her from time to time? Or are they given instructions from the Crown when they're first appointed? Or are they regularly in contact with London?
How about earlier examples such as Roman provincial governors or prefects (Pontius Pilate comes to mind) or Spanish viceroys in Spanish America, or British governors in the Thirteen Colonies or pre-telegraph India?
The role of Governor-Generals in NZ, Australia and Canada changed with the adoption of the Statute of Westminster from a representative of the British Government to a constitutional position within that nation. Even before then, there were heavy tensions between Governors and the Colonial Office. George Grey, of both South Africa and NZ was famous for disregarding imperial policy as much as he could get away with.
In modern times there is no question of them relying own their own judgment and what they believe to be best for the country. Even then, there is a strong convention that Governor Generals will refrain from acting except in cases of crisis where their reserve powers may be required. For example, in 1926 the Governor-General of Canada Lord Byng refused the request of the Prime Minister William Lyon Mackenzie King to dissolve Parliament and call a general election and in 1975 the Governor-General of Australia dismissed the Prime Minister Gough Whitlam. Other instances of the use of reserve powers include Pakistan in 1955, Grenada in 1983 and Fiji in 1987.
Sources:
Laws of New Zealand Constitutional Law (online ed) Constitutional Law of New Zealand, Philip A Joseph, 4th Ed. Making Peoples, James Belich. Constitutional Conventions: The Rules and Forms of Political Accountability Geoffrey Marshall.
Governors-General in Australia use their own judgement. In the famous situation where the Governor-General dismissed a sitting Prime Minister in 1975, he only advised the Queen of his actions after the fact. And, when the Speaker of the soon-to-be-dissolved Parliament wrote to the Queen to get her to reverse her Governor-General's actions, the response from her office was basically "This is none of the Queen's business; it's up to the Governor-General."
It's worth noting that the constitution of Australia is written in such a way that the Queen can't actually exercise her own powers: the constitution specifically says that only the Governor-General can exercise the powers of the Crown on the Crown's behalf (except, of course, for the power to appoint and dismiss a Governor-General!).