Your employee’s payroll taxes (State and Local withholding taxes) are calculated based on their residence location and work location.
The state of Tennessee doesn’t have a state income tax on wages. While, residents of Kentucky must pay federal income taxes, along with payroll taxes for Social Security and Medicare and state income tax on what they earn. With that being said, let’s ensure to set up your employee's tax information for both states so you’ll be able to run payroll.
In your employee's work location, set the income tax as Do Not Withhold. While for the residence address, set it to Withhold taxes.
Also, I’d recommend consulting with your State Tax Agency or a Tax Professional so you’ll be guided with making tax-related decisions.
For additional insight, you may check these articles: