A payroll is a list of employees of a business who are to be paid for their work there. They would need employee's bank information if those payments are sent directly to the employee's bank accounts.
You are going to need their social security number, official identification, and a completed I-9 to verify that they have a legal right to work in the US - if you are indeed in the United States and really have a payroll.
No. A business would need your bank routing number and account number so they can direct deposit your paycheck (money that you've earned by working for them).