The main prosecuting authority in England and Wales is the Crown Prosecution Service (CPS). In Scotland, it is the Prosecutor-Fiscal and in Northern Ireland the Public Prosecution Service. Each of these organisations bring prosecutions on behalf of the state. They are independent of the police, who investigate crimes.
Other specialist agencies also prosecute: for example, the Food Standards Agency, the Health & Safety Executive (UK) and local Trading Standards (consumer issues, including misleading fur labelling). So do local authorities for a range of offences.
Private prosecutions can be a valuable campaigning tool, especially on issues where animal protection organisations have greater expertise than the state authorities or may perhaps have conducted an undercover investigation revealing inadequate regulation.
A private prosecutor has to persuade a magistrate to issue a summons or warrant against the proposed defendant. There must be some evidence indicating that an offence has been committed. The Director of Public Prosecutions (the head of the CPS) can take over a prosecution, and can then discontinue it. In principle, it is possible to judicially review a decision to discontinue, but in practice that would be difficult unless, perhaps, the decision was overtly political.
It is often possible for a private prosecutor to recover at least some legal costs, either from central funds or occasionally a convicted defendant.
It is important to ensure that private prosecutions are done properly, with compelling (admissible) evidence and full disclosure to the defendant, and other procedural requirements followed. If not conducted properly, there is a risk that the defendant may recover costs from the prosecutor. Advocates for Animals can help navigate your way through each step of the process.