I live in Oakwood, Ohio and Work in Xenia, Ohio. My w-2 shows a locality withholding for Xenia. Do I owe Xenia? Is Oakwood tax coming out of my pocket?

I work in Xenia, Ohio and live in Oakwood, Ohio. I want to figure out if my employer made an error with my locality withholding. I want to know if I owe xenia local taxes for working there. I also want to know if I do not owe can I get a refund to apply to the city of Oakwood.