I live in one state but my job has taken me to others and in the past I had employees working in more than one state. The taxation status all comes down to the state involved but few fail to tax non residents however they can. Most have an income limit below which no return need be filed or tax paid (note that if an employer withheld a return must be filed anyway to recover any). Example: Oregon allows non resident employees to make the final call as to whether their employer withholds income tax, but requires a return and potentially taxes if annual income exceeds $2315 (filing singly). In my industry one construction project can trigger that.
The other wild card is the state where the employee is hired regardless of the state where he/she is resident. An Oregon resident hired by a Washington (state) contractor or vise versa adds all kinds of wrinkles depending on where the employer works the employee. So most here are correct; Washington has no income tax (at the moment) but an Oregon resident, even if they work only in Washington, is still liable for full Oregon income tax. If they work in another income tax state as well, typically they can be liable for income tax in both states (subject to the limits), however, to the best of my knowledge, the various states will credit income tax paid to other states so the employee is not taxed twice for the same income.
A good bookkeeper is essential, and often an employee who did not exceed the limit was never aware that tax might be incurred because the bookkeeper was watching.
I yield to any tax expert; I do not claim to be!