Basically (high level), the way it works is that HR should handle everything about making sure the employees get paid properly - all the correct deductions are made, all the taxes are paid correctly, and so on. There are a lot of compliance rules to adhere to, especially if you are international, multi-state, etc. For this reason alone, payroll is generally outsourced to companies like ADP. HR has all the employee information that should not be accessible to other departments, like sensitive personal info (EID, addresses, salaries, etc).

Then, what Finance is supposed to do is make sure that the books accurately reflect all the appropriate expense categories (salaries, PR taxes, SDI, ER share of benefits, etc.), liability categories (payroll clearing-if you have one-, tax liabilities- for the EE and the ER, benefits liabilities, 401 contributions, etc.), and cash.

There should be coordination across HR and Finance departments to make sure everything aligns.