Purchasing a property out of state- taxes

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We reside in NH- a no state income tax state, we want to purchase our next property, a vacation rental, out of state. However, there is state income tax in the states we are looking. How does that work? Do we need to file 2 state tax returns? Our state won’t give back a portion of the taxes paid to the other state since we live in a state that doesn’t collect state income tax.