Whether your mailbox has been bashed or your neighbor is playing music that is disruptive, your responsibility goes beyond calling the police anonymously. If you want something done to correct the situation, you need to be willing to give your name, address and phone number even if all you want done is for a warning to be given. The officer may have more questions for you, or they may be confused on the location of the crime. The police do not have the authority to press charges in all crimes unless you give it to them. Not all of the pieces of the crime puzzle may be known initially. It may take a few days or weeks to get all the pieces together.
On some crimes you probably won't have much luck without the help of your neighbors. If there are no leads in your crime, if you have no clues, the police will be able to write a report that a crime has been committed, however, there is a slim chance of restitution without your neighbors' help to gather more clues. Remember, your neighbors also include the blocks around you. Similar crimes may be occurring in a six block area and unless you know how to contact those other blocks with questions you may not be able to solve your crime. Neighborhood Watch helps by establishing a block to block a phone network.
If you do not file a report and follow through, you allow a crime to happen.