How do you know if you pay taxes on your local School?

Check you W-2 form. The city or locality that you paid the tax to is listed in box 20 and taxes withheld are listed in box 19.

What local school do your think you need to pay income tax to?

If you're asking about Ohio School District Income Tax (SDIT), be advised that 

TurboTax (TT) does not automatically prepare a School District return, you must initiate it.

At the screen, in the Ohio program, titled  "A few things before we wrap up your state taxes"
      scroll down to School District taxes. 
Or later at the screen  "Your 2021 Ohio Taxes are ready for us to check"
    -Click Other situations
    -Scroll down to School District taxes.

Usually  the School District return  is e-filed with the state return, but there are exceptions. Read the Ohio 'filing instructions" that TurboTax prepares, They will tell you if the SD100 was e-filed or if you have to mail it.