Pros of Law Enforcement. The two biggest upsides of working for law enforcement in an official capacity is access to the crime in question. You don't have to answer any, "Why is this character here?" questions, and your characters have access to official case materials and can interact with persons of interest in the case with the authority of being officially involved. In terms of access and ease of integration into the narrative, you can't do much better than a character who has been officially assigned to solve the case as part of their job.

Cons of Law Enforcement. The downside of working for law enforcement is the fact that almost all organizations have specific rules and regulations about how their employees have to operate. And while you can certainly write a character that plays by their own rules, that freedom only goes so far before you have to realistically address the fact that a cop is breaking and entering to investigate, or a federal agent is coercing a confession by beating a suspect. The access to case materials is offset by the fact that law enforcement officers have to go about the investigation in a particular way in order to do their jobs effectively.

Pros of Private Citizens. While they are constrained by the laws that all other citizens adhere to, private citizens do have a certain freedom that allows them to operate in ways official law enforcement can't. They might be able to utilize different resources (hackers, criminal associates, questionably obtained evidence, etc.) than their official counterparts, and their motives for taking a case may have nothing to do with building a solid case that will hold up in court. If, for example, a private investigator is trying to solve a murder, they may simply be trying to identify the culprit rather than working to build a case a prosecutor can use to put the culprit away.

Cons of Private Citizens. The flip side of the coin for private citizens is that they don't have the access that an official law enforcement agent would be given. They may not have access to a forensic lab, or be authorized to bring persons of interest in for questioning, or even have access to the information aggregated by all of the people working on the case in an official capacity. While private citizens can more freely and creatively solve their cases, they also have to contend with the fact that they're operating outside formal channels and may not have the access or the authority that normally compel people to give them the information they need.