It depends. If you perform all of your work in State A, then you should not have to file a non-resident return for work performed for a company that is located in State B. What matters is in what state you earned the income. If all your work was in State A, you do not need to report your income to another state regardless of where your employer is located.
If you performed the work in State B, you will need to file a non-resident return for that state, unless that state has a reciprocal agreement with State A. State A, where you reside, will want you to report all of your income regardless of where earned, but some states give will give credit for taxes paid to another state. States that have reciprocal agreements are where you do not need to file a non-resident return even if you worked in the other state. Please see a list of states that have reciprocal agreements here.
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