If you want a summary...NO, they do not offer reciprocity for any state or NREMT. However, if you hold NREMT and it is current, you may then have your Training Officer/Coordinator submit the training you underwent to obtain your NREMT and if it meets or exceeds the state requirements, the state may choose to grant you a HI license. (You do not need to do this IF you took your EMT class in HI).
I'm in Hawaii (as a Firefighter). We did a full NREMT class as part of our Academy (Honolulu Fire) including ridealongs with EMS and hospital clinicals, and got NR certification.
However, HI state EMT requires additional education hours and requirements above and beyond NR (so we're limited to EMR/First Responder scope of practice as a Dept, even the guys who came in from EMS and are State Licensed...).
There IS talk of there being a bridge course to go from NREMT to State EMT, apparently it comes and goes, and I dont know too much about what it entails, nor if it's only available because we took NREMT in state or available to anyone with NR/out of state certs.... I need to do more research into that myself as I have been thinking about getting state licensed myself.
Just general info, AMR is pretty much the only private ambulance I've seen here. City&County of Honolulu (Oahu) we have our own third service municiple EMS Agency (although guys in the FD who also work part time for AMR say roughly a quarter of their calls are backup 911 for the City, the rest are BLS, ALS, CCT transfers, although they're first in at the airport as well).
(Edit to add, seems fairly related to the topic:
You need to email Kapiloani Community College (KCC). The course is EMT 135 it allows NREMT medics to gain the required ride time for Hawaii and you are required to have ITLS since if you read the documents for the State of Hawaii our "EMT" are according to the State of Hawaii licensed at the AEMT level. You also need an IV course and training on the monitor as well. Email the head of the paramedic program at KCC. While reciprocity is not offered directly if you go through the transition process an NREMT EMS provider can gain a Hawaii Cert.