I did some reading at NC Dept of Revenue and searched around for info.
My understanding is that, in any case, the fiduciary (and not the beneficiary) has the responsibility for reporting any applicable income to NC. I'm reading the instructions for form D407A. It looks like the beneficiary is only required to report whatever distributions are on the K-1, as usual.
Also, see "Kimberly Rice Kaestner 1992 Family Trust vs. North Carolina Department of Revenue". The law that says out of state trusts have to pay tax to NC because a beneficiary lives in NC was decided to be unconstitutional by the Wake County NC Superior Court, but only for that one case. NC DOR is expected to appeal.