The police won't come looking for you unless you use the information for illegal means. For example, you have a man who abuses his wife and threatens to kill her. She leaves him and runs to florida and changes her name and loses her 'old' identity. The husband then hires a private investigator to find out where the wife is. He finds her and gives that information to the husband....and then the husband travels to Florida and kills the wife. A crime was committed so the husband will be looked at (obviously) and so will the investigator.
I've researched old friends, etc and have been able to find information on them on-line. There was also a woman that I was involved in a lawsuit with and I found out a lot financial information out about her, such as the value of her house, cars, employement, etc. It was all very easy to obtain and I think it cost me like 80 bucks.