I'll just sum up my answer to the original question by saying that cops have authority over you because they have decided that they respect and believe in the law enough to become trained and certifed to uphold it according to standards set by the people who we have elected to make laws. They feel that it's important enough to dedicate their lives to it.
BTW, every teacher or professor who I have asked has replied that the concept of a citizen's arrest is entirely fictional, and that all you can do is hold a person in place until the police get there. Theoretically, you can thorw them in the back of your car, read them their miranda rights, and drag them down to the station, write and sign a statement about what you saw them doing, but the police still have to do the paperwork and official procedures, and the guy isn't going to be too happy about being thrown in the back of a car by some random person and dragged down to the station.
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