Australia as far as I know
Master of Philosophy (Higher Degree by Research)
It is important for you to check the website of the universities/institutions where the professors are working. All the information about scholarships will be available there. If you just email them without looking for information yourself, you are wasting their time.
Probably, there will be scholarships designated for specific projects which professors are looking for students. You may contact them to express your interest, and to ask them if they are willing to supervise you. Then you can proceed with application for the programs and the scholarships, yet it is worth mentioning that these scholarships are mostly for PhD students.
For general scholarships (many universities offer Master by Research/MPhil scholarships without specifying the eligible programs and projects), in my opinions, it is not appropriate to send emails to potential supervisors solely to ask for Master (by Research) scholarships. You may send emails to them to express your interest, to know more about them, and to know if they are willing to supervise you. If they agree, you (and the potential supervisor(s)) can proceed with application for the programs and the scholarships.
Professors are generally not in charge of giving scholarships even if there will be designated scholarships for potential MPhil/PhD students working on their projects, yet the support of potential supervisors will influence the chance of getting the scholarships, so the most important step is to convince them to supervise you first. Without this, no scholarship whatsover.
Moreover, make sure that you are eligible to apply for the programs and the scholarships.
Master by Coursework
If you are not interested in doing research, and you simply want to apply for a taught master, there will be probably no full (merit-based) scholarship from most universities. Probably there will be 50% scholarships from some (good) universities, but the required GPA is really high (min: 6.5 - 6.8/7). 15-30% scholarships are probably more common (min: 5.5-6/7), but given the high tuition fee and living cost in Australia, I think you already have the answer for this.
For most universities, you can get at most ONE merit-based scholarship. There will be top-up scholarships that are worth 500 to a couple of thousands of AUD. Depending on the universities you are applying to, you can get a few top-up scholarships, but there will be a limit.
Professors will probably have ZERO influence on these scholarships mentioned above. You just need a good (or more precisely, a ridiculously good) GPA.
There is a prestigious scholarship from the Australian Government (the Australia Award) which covers everything, but you have to go back to your home country and stay there for at least 2 years? after finishing the degree program unless you repay every buck they give you. That is a limitation as you cannot apply to any working VISA after graduation (e.g 485), and cannot leave your country for a long period of time during these 2 years, but that is the purpose of that award which is to help developing countries. If you do not care about working opportunities in Australia after graduation, or you have no plan for further study after graduation, I think it could be a good opportunity. Just do not take advantage of it for unethical motivations.