I spend a good bit of time on the Big Island every year for R&R, but usually play little golf. Of the ones I've seen or played, though, would rate them as follows (no differentation of private/public). Also, my criteria is fun, not rankings, as this is why you play vacation golf.
Nanea, Mauna Kea, Mauna Lani (Ocean), Makalei, Big Island, Hualalai.
I thought Nanea was pretty interesting. Doesnt' look like much from the road (golf on a steady sideslope) but many more subleties on site that Kidd and his talented crew (Jay, Paul) did a good job with.
Heard Mauna Kea was looking more encroached of late with mansions. Agree it's not really up to the ratings it's gotten, but still plenty of good and fun holes...would say more than just #3, certainly.
Just played Makalei. Similar to Big Island, in that it's "inland", up at an elevation of 2-3,000 feet or so, core golf, Dye group (I think) and diverse in it's vegetation from what you're used to down by the water. Felt a bit like Jurasic Park! Is 900' of elevation change, which amazed me (and they display it on the card like that's a good thing!). Of course, cart-golf, but had some really fun holes. I liked it better than Big Island, in that it had fewer railroad tie-lakes, and better shaping. Is in worse shape, though. Intrawest took over Big Island, and supposedly in good shape now.
Need to see Hokulia (new Nicklaus), as I've heard it's good, and heard from Kidd group that Waikaloa Kings course was interesting, but haven't seen it.
Have fun! That's what I do there. It's not the best golf in the world, but can be very enjoyable.