Otago has somewhere between 290 and 300. Not sure about Auckland.
Although it varies from year to year, ~75% of those places are for undergraduate entrants. ~20% are graduates, and ~5% are for others. Nobody knows how many people actually apply except the admissions office. Although, you must get >70% in every paper (B grade), so anybody who enrols into HSFY and scores below that will immediately be excluded from entry. There are those who sit HSFY papers but are not intending on applying to a professional course (eg BSc students). There are many people who do HSFY but are intending to enter a different professional course to medicine. I would say there are ~1600 people who begin HSFY +/-10%, but probably greater than half of that number either switch to something else, don't make the B grade, or do not apply for medicine.
Otago also have a greater number of places than Auckland that tend to be offered in accelerated entrance schemes: together, we have more Māori, Pacific and Rural students. Accelerated entrance applies across all three entrance pathways. This does make entrance more competitive for general entry, and admission rates for Māori and Pacific are increasing as their academic performance has been greatly increasing over both undergrad and postgrad.
I hope this helps. Much of the numbers I've said are purely speculation, but it's the general impression I've been given over my 5 years down here
-LM