Q. Does the PA-OH agreement apply to local taxes too?
A. No.
Q. How does it work with local taxes?
A. The PA city is allowed to withhold for the city you work in (your employer is required to do that). You cannot get that refunded. But, you do not need to file a PA city tax return.
Q. Do I need to pay both PA local taxes and RITA (Ohio) local taxes?
A. Yes. But your Ohio home city MAY allow a credit, or partial credit, on your RITA return, for the tax paid to the PA city.
List of credits at https://www.ritaohio.com/TaxRatesTable