pitcairn 01
Pitcairn Island has just 45 inhabitants or so, and the police/customs/immigration/agriculture officer is just one person.  Meralda Warren performed these functions for a long time, and much information is available on the Internet about her.  She even has her own website selling handicrafts from the Island.  Whoever has this position and wears these many hats does not wear a uniform.  Therefore, these patches were never purchased nor used by them.
You should treat them as FANTASY items made to deceive collectors into obtaining something great and rare, yet with no basis of legitimate use.
PN 001 Customs and Immigration Department Type II
Pitcairn Islands - variation of above patch, perhaps another batch of patches has been produced....
Roy Stammwitz collection 
pitcairn 02
Folks who show up on the Island are directed to the officer within 24 hours of arrival, and the officer stamps the bearer's passport after a short interview.
Stays are long as boats are infrequent, and the arrival of visitors is known well ahead of time.  There are forms a potential visitor must complete before setting foot on the Island.
Courtesy of Mike Pirnie, 2009 for the above information