In New Zealand, a Notary Public is appointed by the Arch Bishop of Canterbury.
The powers and functions of a Notary are generally applied for international purposes. Documents for use overseas are required to be authenticated by a Notary. For instance, if you are applying for a teaching position in say United Arab Emirates, you will be required to submit notarised copies of your qualifications.
Similarly, if you are a beneficiary in an estate which is administered in USA, you will be required to provide evidence of your identity. Such evidence will normally take the form of an affidavit which must be signed before a Notary.
An appointment is required for notarising documents. Notaries are entitled to charge for their service and proof of identity is required either by a current passport or other photo identity e.g. drivers licence.