Sorry, JLTmsu! I haven't checked back in a while! But just to try to answer your questions...
For "beyond" I mean for residency and registrar or specialty training (e.g. like if you want to become a surgeon or whatever). Basically, if you aren't a NZ or Aussie PR (permanent resident) or citizen, then it'll be really tough to get an internship in NZ or Australia. Internships and RMO jobs and so forth are given to locals first, as far as I'm aware. Also, internships etc are given to people from a medical school in the same state, at least in Australia. So if you are from New South Wales, then NSW locals (i.e. PRs and citizens), who did high school in the state, and who did med school in the state, would be preferenced over interstate applicants like from Victoria or Queensland for example. That's how I understnad it at least.
I'm not sure how citizenship works in NZ? But for Australia as I understand it, it'd depend on what pathway you're applying for citizenship (e.g. via marriage, via a needed skill). I might be wrong and others can correct me, but I don't think an international student studying medicine without other arrangements like marriage to an Aussie citizen or a skill could get citizenship easily though, at least not until after graduation once you're a doctor. But then that again makes it hard to get an internship because you won't be a PR or citizen unfortunately. 🙁