Anyone know how fingerprinting kits work?

You'll have to follow some links from here, but it looks promising: http://en.wikipedia.org/wiki/Fingerprint

Our bodies secret sweat and oils. Each person has absolutely individual raised ridges on their hands and feet. The most clearly defined patterns are on the fingers and toes, and so those are generally used for identification. Whenever you touch an object some of the oil/sweat is deposited on the object. Since our toes are generally not used for touching, fingerprints are the primary prints we leave behind us. These are called latent prints.

The simplest means of lifting a print is to lightly dust the printed area with a fine coat of black powder. A large soft brush is used to evenly distribute the powder. A good technician will whirl the brush just over the surface to avoid disturbing any potential Latent prints. When the technician finds a print, they carefully "lift" it by covering it with a clear tape, and then putting the tape onto a card. Lift cards have a printed form to identify the day, date, time, case number, precise location of the lift, the lift surface and the technician's name and number. The lift cards, there are usually more than one, are then entered into evidence along with other physical evidence from the scene.

Back in the "lab" the fingerprint examiner, who may be the technician or a specialist in analyzing prints, will figure out which digit the lift came from. The examiner will then classify the print according to its characteristics. The classification is recorded on the lift card along with the examiner's name and number. The examiner then enters the classification information into the local data base, and returns the lift card to evidence.

As you might imagine there are millions of rolled and lifted fingerprint cards in even relatively small urban jurisdictions. In the past it was virtually impossible to match a lifted print to a rolled print unless you knew who the print probably belonged to. A match comparison would be run against all the known people whose prints would be expected to be found at the scene. By temporarily eliminating those lifted prints, one hopes to isolate and identify the suspect(s). As the investigation proceeds and suspects are identified their prints would be compared with the lifted latent prints from the scene. This was a powerful tool for criminal investigators throughout most of the 20th century, but the process was very labor intensive and often proved of limited value. Home owners who were burgled expected their homes to be fingerprinted and so many agencies would go through the motions, but such prints were usually sloppy and very rarely useful in building a successful case. Public locations where hundreds of people leave their prints were also usually useless.

Prime locations to search for latents are around doors, windows, drawers, and light switches. Of course, criminals quickly learned to wear gloves and/or wipe down surfaces that they may have touched. Sometimes they get careless and investigators will get lucky. I've known criminals to be convicted on the latent fingerprints they left on bullet casings left at a scene where no other prints turned up; they handled and loaded a pistol with their bare hands, and then were careful to leave no other prints behind. That is still considered luck, even though one of an investigator's best tools is the criminals own stupidity.

Some surfaces will capture really great impressions, and others present great difficulties ... especially for traditional lift methods. Glass and other hard polished surfaces tend to be good latent locations. Cloth, rough surfaces, and skin almost never yield useful prints by traditional methods.

By the late 1970's a revolution in fingerprints was getting started. New methods of finding and lifting latents were discovered and developed. Technicians began using special lights that are able to reveal the presence of latent prints without the use of fingerprint power. Smaller objects could be isolated in a chamber, subjected to fumes that "developed" the prints, and cause them to slightly fluoresce. This, and other new treatments, made it possible to recover useful prints from surfaces that earlier were not possible. A detective in California (Det. James Mock) developed these techniques, wrote a book on the subject, and was for a number of years in great demand as other agencies around the world learned the new methods. Many other methods have since been discovered and are in at least partial use today.

The second factor in revolutionizing fingerprints came with the power of computerization and communications. Computers can build really huge databases, and search them for relevance almost instantaneously. The earliest databases consisted of just the classifications made by examiners, but since examiners can easily make mistakes that was found to be mostly inadequate. Software was developed to capture the exact image of the rolled and latent prints, and enter those into the database. This sounds simple, but it isn't. Often the latents are distorted or only partially captured. Comparing latents to rolled prints was a problem until better software was able to adjust the distortions. This is the AFIS system and its come a long, long way from its inception back in the 1980s. Early on many agencies began to use the system to clear old cold cases with a great deal of success. Unsolved crimes of over 20 years were suddenly "solved" by matching old latents to rolled print databases. The problem was that the system quickly became swamped in fingerprint inquiries, and most of the cold case efforts had to be curtailed to handle fresh cases.

The system is now pretty much standardized. Modern communications makes it possible to link databases in virtually real time. The largest data banks are, of course, in the FBI and Federal data banks. Local agencies almost routinely now query those databases that are available, one by one. Federal investigators had the same problem, and a print in one system couldn't be found unless the investigator had access. Federal agents generally do a better and more complete job of searching national data bases because they have better access. A local sheriff, for instance, will almost certainly not have access to many Federal data bases. On the other hand, there are so many local databases that Federal agents can't efficiently search for latents that may only exist in a few small jurisdictions.

Then 9-11 happened and finally an effort has been made to unify and link databases. I wish I could say that has resolved the problems, but it hasn't, nor is it likely to in the near future. Modern fingerprinting has become a fundamental means of identifying suspects, but it isn't perfect and no one should expect it to be.

I hope that fills your need and answers your basic question. BTW, private detectives haven't a prayer of using fingerprint evidence and I greatly doubt that any even try to use fingerprints for any purpose. Private eyes just don't have the resources available to work most criminal cases, but they do have the time to persist where public agencies are under constant pressure to resolve a case and get on to the avalanche of new cases that stack up everyday.