Every organisation needs a privacy officer
The Privacy Commissioner's guidance is clear that every agency covered by the Privacy Act is required to have a privacy officer. An agency here means almost any business or organisation, not just a government department. The Commissioner describes the role as ensuring the agency meets its obligations under the Act, dealing with requests from people to see or correct their personal information, and working with the Commissioner during the investigation of complaints.
For a board, the useful questions are simple. Who is our privacy officer? Do they have the time, authority and access to do the job? When did they last report to us, and what did they say? In a small business the privacy officer may be the owner or the office manager. That can work well, as long as someone has actually been given the role rather than assuming it belongs to someone else.