I would be most surprised if they cared where you actually stayed in the end. No-one is actually going to check the police records against the visa application records! Of course you can inform the consulate as Often1 suggests to give you peace of mind, but maybe they'll then ask you to come in and submit other proof....Consulates, their staff and their pprocesses are (generally) not 'designed / trained' to be especially user-friendly.

You dont need to actually contact the local police. When you check into the hotel, they'll ask for your passport and (usually) keep it for a few hours. They enter the details into a register, which goes to the local police. In smaller / older hotels, it is all done manually on pre-printed registers, maybe it is different in the newer / bigger hotels. My latest experience in Italy is 2 years old.