Well, if management is competent, with an interest in how the people it's employed, work together, it'll already know. And if it is not competent, and has little interest in how the people it's employed work together, then reporting to it is at best futile, and will at worst lead to firing of the immediately identifiable voice of disharmony.
I for one would not report a coworker. It undermines the trust and cooperation necessary between me and my (other) colleagues, and it's a responsibility my boss, not I, is being paid to shoulder. Not to mention that, at some point, I'll make a mistake as well, and might wish to explain it myself rather than being told upon before I've had any chance to correct it and decide upon a strategy to avoid repeating it.