You do not need to report anything in Ohio.

Since Ohio and Kentucky have a reciprocity agreement, you do not need to report anything for Ohio. 

You will report all of your income in Kentucky.  Your W-2 form should have Kentucky listed as the state wages and withholding in boxes 15-17.

If your W-2 form shows withholding for Ohio, you should submit the attached form to your employer requesting that they only withhold taxes for Kentucky.

Ohio reciprocity

If your W-2 form does show OH, please comment and we can assist you with your input.  If it does not, you only need to file your Kentucky return.

On my Ohio W-2s line 15 says Ky and shows state wages and state income amounts, but on the other line 15 shows Oh with local wages and local income tax. I'm confused.

Some cities and villages in Ohio have local taxes.  This should not be confused with state taxes.  Generally, there is a tax where you work.  This is a flat tax based on your wages.  You do not have to file a local income tax return if your employer withheld the correct amount.  You will neither owe anymore nor receive a refund.