You'll need to make sure you're a New Zealand tax resident. You do this by working out your tax residency status. This is different from your immigration status.
In general, New Zealand tax residents pay tax to New Zealand on their worldwide income whether or not it is sent to New Zealand or been taxed in the other country or territory. New Zealand may give a tax credit for tax paid overseas.
If you're a new tax resident or returning to New Zealand after 10 years, you may be eligible for a tax exemption on most types of foreign income for approximately 4 years.
If you earn income in a country or territory which has a double tax agreement with New Zealand, this may affect how you are taxed.
International tax can be complex and you may want to consult a tax professional if you're in doubt about your circumstances.