> Can it be done?
Yes, but.
It is fairly easy to work in NZ or AU for a set period of time once you have your primary medical qualification, it is far more difficult to work there for good.
> What hoops do I have to jump through?
Here are my five cent (I have no personal experience, I know however a couple of people who did either sabatticals DU or worked there for a while). I have toyed with the idea of going to NZ for a year or so, I hear it is a nice piece of earth and all of the NZ docs I have met here in the states so far couldn't wait to go back, the better financial opportunities in the US nonwithstanding)
NZ
Once you have your US board certification, you can go to the NZ registration board and have your credentials evaluated. If they find them substantially comparable to the NZ training, you can get a 'vocational registration' to practice limited to your specialty. These positions are pretty much limited to the public health sector (which from my understanding provides the majority of medical care in NZ).
I don't know whether there is any reasonable access to training positions in NZ for foreigners. Judging from the experiences of the NZ docs I have met doing their residencies in the US, access to specialty training positions is competitive.
NZ accepts USMLE for access into temporary and training positions. If you get your specialty certification recognized for a 'vocational registration', you don't need to write their local licensing exam NZREX.
AU
Through restrictive and idiotic immigration policies in the recent past, AU has a considerable shortage of physicians.
In order to practice independently in AU as a foreign trained physician (sorry, overseas trained (OTD)), you need to write the AMC (australian medical exam) and wait for 10 years until you are eligible for a medicare billing number.
However, in the meantime the aussies are running out of physicians. Even in the outer suburban belt of major cities, positions in public hospitals remain unfilled (the positions are thus designated to be in an 'Area of Need')
This seems to be mainly at the attending level. For these so called 'Area of Need' positions, AU imports foreign physicians after sometimes only cursory check of their credentials (there has been a recent scandal where a surgeon who had lost his license in NY and got it restricted in Oregon managed to finagle an 'Area of Need' registration in Queensland. Allegedly he butchered a couple of esophagus resections and hurt trauma patients by refusing to transfer them. When the medical board and parliament investigated him, he skipped the country and is hiding in Portland, OR. It is a whole political storm in Qld, endangering the future of the AoN program.)
But even at the level of employed junior physicians for inpatient care, there is a shortage. Positions as 'resident medical officer' (RMO) are readily available and there are agencies actively recruiting residents in europe and South Africa. However, THESE POSITIONS ARE NOT ELIGIBLE AS TRAINING POSITIONS. In order to become eligible for the respective australian college exams, you need an accredited training position. And in order to get these training positions, you need to be a permanent resident of AU and you need the AMC (with some exceptions). Also, access into the AU accredited specialty training positions is competitive. (Often, you have to write the first part of the board exam in the respective specialty, just to be eligible to apply for a position.)
So, if you really intend to stay in AU for the long haul, you might want to think about writing the AMC up front. It is from my understanding not a trivial exam, and you might think about writing it alongside with your USMLE's while the subject matter is fresh in your mind.