Unless your state has a tax agreement with Ohio, you would have your income taxes withheld in Ohio. You would file an income tax return for Ohio and your home state. Your home state will normally allow a credit for taxes paid to Ohio - usually up to the amount you would have paid to your home state on your Ohio income.
Example: I live in New York. If I work in Ohio, I would file a non-resident return to Ohio and a resident return in NY. NY will tax me on all my income, but Ohio only on the income I earned there. NY gives me a credit for the tax I pay to Ohio. However, if I work in a state with a higher tax rate - like Massachusetts - NY will only allow a credit for the amount of tax that I would have paid on the MA income in NY. Since MA has a higher tax rate, NY will not credit me for all the tax I pay to MA - just for the amount of tax that I paid to MA that is up to the amount I would have paid to NY on that same income.
In my working career, I have filed income tax returns to as many as seven states in the same year. Every state is different. Many do not require you to file a return at all if you did not earn above the threshold for filing. However, in order to get any refund you may have coming, or to claim a credit in your home state for taxes paid elsewhere, you will have to file a return anyway.
The only way you can avoid having your Ohio taxes withheld in Ohio is if your state and Ohio have a mutual agreement - like the one between NY and NJ - which would allow your taxes to be withheld in you home state. Very often, states which border each other have such agreements.