Filing taxes as a student

Q, Should I file taxes in both states even though I have no income to report in either?

A. Simple answer: No.  Federal and state income tax filing requirements are based on the amount of income that you have.  Since you have "no income" you do not file in either state. 

You say you "receive state-funded assistance in my home state and school's state".  Depending on the nature of that funding, some of it may be taxable income.  If scholarships and grants, exceed qualified educational expenses, the excess should be treated as  income. Qualified educational expenses (QEE) are tuition, fees and undergraduate course materials).  Essentially scholarships that pay for room & board are taxable. But if that is your only income, and the taxable amount is less than $12,950, it does not have to be reported.

Students, who attend school out of state, are still considered residents of their home state, unless they take concrete steps to change residency.  Taxable scholarship would only be taxable in your home state, not the school state.  Income from a  job, in the school state, would usually be treated differently.

It's not clear what " live about" means.

You only have one permanent residence.  Usually for college students, that is your parents' home, because you have a reasonable expectation of returning there after graduation, at least until you figure out the next steps in your life.  It is possible, though uncommon, for a college student to establish a bona fide new permanent residence someplace else.

As a permanent resident of state A, you are required to file a tax return for state A that reports all your worldwide income.  As a temporary visitor in state B, you would file a non-resident return for state B that only reports income earned while living in state B.  State A will give you a credit if you pay out of state taxes on the same income twice.  So suppose you have investments, a summer job in state A, and a school season job in state B.  You would pay tax in state B for your state B job only, you would pay tax in state A on all your income (investments and both jobs), and then state A would give you a credit for the tax you paid in state B on the state B job. 

But of course, if you have no taxable income, you don't file a tax return and you won't pay taxes in either state.