This does depend. At a basic level you are legally responsible for your own personal data and anything that happens in your own name. This means that if a criminal uses your name to commit a crime or run up fraudulent charges you will have to first prove it was not you that carried out these actions, but someone else using your stolen identity before you can clear yourself of any legal responsibility.
If a criminal has committed a crime in your name will need to hire an attorney as soon as possible to help you through the process of getting the charges cleared, as not only will this mean you have a false charge on your criminal record until the matter is resolved which could prevent you from applying for jobs and credit, but you may also end up being repeatedly detained by law enforcement officers who may have outstanding warrants in your name.
If your identity is stolen you must report it as soon as possible to the police, your credit account providers and the main credit reference bureaus, because if you fail to notify all of the relevant interested parties in a timely manner they may well hold you responsible for any later charges run up by the thieves, and you will end up paying for them one way or another.
Also your account providers will expect you to protect your own account information and keep it private and secure, so if you were the one who revealed your information (such as credit card numbers or online account logins) to another person who then went on to misuse your data, you will most likely be held responsible and have to pay some or all of the charges and debts run up in your name.