Well, before the usual "depends on what kind of story you want" type of advice, we may as well just list the pros and cons of each.
For Private Eye, we have a greater sense of freedom, at least three different archetypes to choose from, a greater potential for diverse locations and types of cases, and generally more flexibility when it comes to the story itself, since there's no strict legal protocol to follow. However, there are certain methods private eyes aren't legally able to use, such as search warrants - if anything, they may be charged with obstruction of justice if they go breaking and entering. Still, they can be hired as consultants directly assisting the police, though this would typically require some particular expertise uncommon in your average forensic unit.
For Police Detectives, we have a much more thorough ability to apply the law on-site, using information and resources not easily available to civilians. However, there are procedures to be followed, red tape to go through, and there's a limit to how diverse the cases and locations can be. A Private Eye may travel the whole world for any type of case, but a city cop is a city cop, and working in homicide doesn't often lend itself to cases related with art theft or kidnapping etc.
For a shared universe, assuming the other characters are not of the police or other emergency services, I'd recommend going with the Private Eye. It's an enduring and flexible archetype with plenty of creative options to meet your needs, and can easily mesh well with all kinds of characters and be useful in just as diverse a range of stories.
A private detective also has the benefit of a confidentiality agreement with his clients, similar to those of a doctor, a lawyer, or a priest taking confession. He doesn't have to reveal information to the police, unless they subpoena him for some specific case. A police detective who suppresses evidence would get canned instantly.
This Space Intentionally Left Blank.The setting is going to (for the most part) be a Space Opera / Space Western, with the mystery series being the closest to, if not in, the present day. We want it to be in an Urban setting.

A friend and I are in the early planning stages of a Shared Universe that will include a mystery series. We were wondering which has more potential for storytelling: a Private Eye or a Police Detective? Any help would be much appreciated.
edited 13th Feb '18 12:44:45 PM by HumanTorch2