International Student Funding for Graduate Programs outside North America
I'm a PhD candidate at the University of Western Australia, a Go8 university which has similar application processes to the Australian National University and the University of Melbourne.

In Australia, funding for international students comes from three separate streams. 

The federal government (heretoforth referred to as the Commonwealth) funds several recurring 3 (minimum) to 4 (maximum) year scholarships each year per university depending on total research output. These scholarships are then allotted to the highest ranking applicants for PhD candidature from the respective faculties in a given university. These scholarships are referred to as IPRS (International Postgraduate Research Scholarships) and are extremely competitive. 

The Commonwealth also funds scholarships for students from developing countries through the foreign aid program. These are referred to as Australia Award scholarships and Prime Ministers scholarships. The application process is run through the Department of Foreign Affairs and Trade website (DFAT).

Individual universities offer university scholarships to applicants and these are usually part and parcel of the application process for PhD candidature. These are highly competitive and based on a ranking process through the entire university and the top (x) candidates who do not receive an IPRS are given these. 

The third (and most common) funding stream is via the research funding obtained by individual academics. Depending on how keen a particular grant holding academic is to bring you into his group, he may offer to pay your scholarships out of his own research grant. Australian academics are very used to this so do not be afraid of asking for an assurance from your prospective supervisor that he is willing to pay your scholarship if you are unsuccessful at obtaining an IPRS or a university scholarship. 

If you want to do a masters, you will have to pay for it yourself. Hope this was useful.

Cheers