People have been found to be positive on their tests for leaving the island. I think this is why they want it done officially so they can then contact trace and try not to get community spread. If done unofficially would that person be responsible enough to say they were positive? They want as much control as they can to keep every one safe.

From their press release yesterday:
COVID-19 testing is one of the main strategies employed by the Government of Anguilla in order to identify sick persons early, so that they can be isolated and be treated as necessary. This lessens the likelihood of the spread of the disease on the island.
In light of the importance of testing and tracing of contacts of infected persons, as well as to ensure that all test results are entered into the national data base for international reporting and decision-making purposes, individuals and unapproved institutions are prohibited from importing COVID-19 tests of any type. This includes, but not limited to, RT-PCR, Antigen, Antibody, rapid and saliva tests of any kind, as well as self- administered kits available commercially or approved for research purposes. Regulations are in place that authorize the Ministry of Health to confiscate COVID-19 tests that are imported by individuals and unapproved institutions either for personal use or for sale.
Persons that contravene the regulations are liable upon summary conviction for a fine of $15,000.00 for importation of COVID-19 tests either for personal use or for commercial sale. Additionally, unapproved persons are liable upon summary conviction for a fine of $50,000.00 and a fine of six (6) months if they offer COVID-19 tests for sale to the public.