Shadowing is a requirement for some schools, some even require specifically a letter from a DO. To get that, you'll probably have to travel down the states to get a letter, although most DOs are pretty cool with it, they know what you are after and will gladly write a letter after a shadowing visit or two.
The requirements are similar, you need certain pre-reqs for some (basic science courses), I suggest going to the DO part of this forum and checking out the requirements yourself. They are expensive as well, usually cost just as much or very similar to attending USyd. Life expenses may differ obviously as DO schools tend to be a bit more regional/rural.
I am not going to write an essay on the differences between an MD and DO. Obviously the name implies they are differences, you end up with a DO degree and not an MD. There are certain stimgas that may or may not be attached to that, depending on how you look at it. DOs tend to slightly favor primary care specialities more. There is an osteopathic component, that is basically in the name, so you have to be OK with that. Other things also apply, but you basically end up with the same privledges as an MD degree.