Mr.Everest, you are right on rate , and payroll department or companies are responsible for appropriate withheld from employees too. . They have to withheld the federal , social security and medicare according to the employee income level and most of the case it will closer to their tax liability and depending on the w-2 form fillled by empoyee but if the tax payer works two different places and still holding @15 % tax bracket  but the income are different tax bracket  than withhled will end up more income but less withheld. Result you will owe to government.
If emplyer had witheld two state , to get the money back from the state you are not residing , you must file tax for refund, That will be the easiest way.