That's a good question.
When someone's identity is stolen, one of the most obvious signs is that money is withdrawn from bank accounts or charges are made to credit cards.
What happens if they are opening credit cards, receiving medical care and opening other accounts in your name?
Check all your accounts regularly. If you see any strange activity, call your bank or the company you are dealing with. Check your statements for bills, make sure it's for your account and all information is correct. If you receive a past due notice and it's not for an account or service you received, don't assume it's an error. Check it out and be sure someone isn't using your information. Check your credit report routinely to be sure things aren't showing up there that should not be.
You can learn more about identity theft from a site by the FTC if you are in the US.