Yes, you may need to file in both states if there is Kentucky state taxes taken out of your paycheck. if so, you should file a Kentucky State tax return as a non-resident to receive a refund for the taxes withheld or receive a credit on your Ohio resident return for the taxes paid in Kentucky. If you have no taxes withheld for the state of Kentucky, then you are not obligated to file a Kentucky non-resident return.
As a FYI, Kentucky and Ohio have reciprocal state agreements where there will not be taxes withheld in those states for non-residents. If this is the case, you will need to fill out a form 42A809 and submit to your employer. Here is a link for that form.
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