The short answer to this question is ‘no’. There are different types of visas which allow non-nationals to work legally in Australia: some require employer sponsorship, others do not.
The temporary skilled work visa (also known as a 457 visa) is one which does require an employer sponsor, and is probably the most common type of work sponsorship for Australia. This visa allows holders to live and work in Australia for up to four years (which can then be extended for another four years), and is available to skilled workers across a range of professions. The idea behind this scheme is to allow people with specific skills which are needed by Australian businesses to fill those gaps, and requires an approved employer to confirm that the applicant has skills and/or experience which are necessary for their business.
Under this visa, immediate family members can accompany the visa holder for the four year period, and visa holders can exit and re-enter Australia as often as they like. Dependents, spouses or de facto partners of 457 visa holders can work or study in Australia without needing their own visa. In fact, their situation is actually more flexible than the primary 457 visa holder as they can change employment, and work and study wherever they choose, whereas the original visa holder must remain with the company who sponsored them, in that specific position, in order for their visa to remain valid. To secure a Temporary Skilled Work/457 Visa, applicants need to apply for the 457 visa once an employer has secured approval from the Department of Immigration, and nominated the specific position. They will also need to satisfy a number of requirements, such as having the relevant skills, satisfying character requirements, and speaking vocational English, among others.
Other types of sponsored visas include sponsored training visas, which allow visa holders to participate in workplace-based training programs or apprenticeships. Some of these visas are restricted to programs in regional areas only.
In terms of visas which do not require employer sponsorship, one of the most common is the working holiday (417) visa. This visa is only available to people aged between 18 and 30 years of age, although there have been some reports recently that the Department of Immigration may revise this to 34. Additional requirements include that applicants hold citizenship of a country which has a working holiday agreement with Australia (including many European countries, Hong Kong and Japan), and cannot have any dependents. The working holiday visa is valid for 12 months from arrival to Australia, and allows visa holders to have any kind of temporary or casual employment, but they can only work up to six months with the same employer. The working holiday visa can be extended for an additional 12 months, through additional restrictions, including an element of working in a regional area, apply.
Those from countries which do not have a working holiday agreement with Australia may also be eligible for the work and holiday (462) visa. This has similar requirements to the 417 visa, but also requires applicants to hold a university degree, and again is only open to applicants from certain countries, including the USA.
In terms of skilled workers, another option to secure a work permit for Australia without employer sponsorship is the skilled migrant independent (189) visa. Although not linked to a specific employer or position, applicants do need to express intent to work within a certain occupation. They also need to demonstrate that they are skilled in one of the areas which the Australian Government has listed as eligible occupations.
More information on visa types and requirements, as well as application procedures, can be found on the Australian Government’s website.