No, the Post Office doesn't have its own "law enforcement" (if you mean something like a police force) or its own judges.
Any company has the right to bring a private prosecution under UK law, and this was the basis for the prosecutions in question. It just means that the company pays for some of the costs involved.
Whether or not private prosecutions should be allowed is certainly a legitimate topic of discussion. Let's not muddy the waters with misinformation about the Post Office having some kind of parallel police and courts system. It just doesn't.
