This is probably one of those questions that got answered back when the 3.x Dragonsong series was still brand new, but I didn't think of it until very recently (and I wasn't around until 3.4 anyway), and I'm just now starting the 3.x series of MSQ.
Essentially, how do dragon eyes work? More specifically, how much influence does a dragon have over their eyes after they've been removed?
I might be mistaken, and please do point it out if I am, but as far as I know, the timeline kind of goes like this:
The Dragonsong war started about a thousand years ago (circa 550 of the Sixth Astral Era, according to the lorebook). King Thordan and his knights killed Ratatoskr and ate her eyes, and gouged out both of Nidhogg's eyes as well. Haldrath is remorseful, and takes one of Nidhogg's eyes, leaving the other in the care of the other knights, who become the ruling class of Ishgard.
So Nidhogg Eye 1 is with Haldrath, and Nidhogg Eye 2 is with Ishgard. Presumably Nidhogg Eye 1 is the one Estinien carries around in the present day, as a plot token in the 30-50 Dragoon Job Quests, and the Heavensward MSQ.
Nidhogg Eye 2 is then revealed in the Singularity Reactor cutscene with Archbishop Thordan and his knights, having been interred with Haldrath's body. I accept that this may in fact be Nidhogg Eye 1, and Estinien has Nidhogg Eye 2, but the general point remains: as of that moment, both of Nidhogg's eyes are accounted for.
Rewinding time a bit: after the fight with King Thordan a thousand years go, an eyeless Nidhogg limps to Hraesvelgr, and guilt-trips Hraesvelgr into giving Nidhogg one of Hraesvelgr's eyes. Hraesvelgr agrees, and so Nidhogg gains Hraesvelgr Eye 1.
Which means for the thousand years of the Dragonsong War, up until the Warrior of Light comes onto the scene, Nidhogg has been terrorizing Ishgard and stewing in his own hatred while wearing Hraesvelgr Eye 1.
Back in the present day, the WoL and Estinien storm the Aery and slay Nidhogg, recovering Hraesvelgr Eye 1. This is returned to Hraesvelgr.
The rest of the MSQ happens, and Estinien picks up the discarded Nidhogg Eye 2 in the Singularity Reactor. Since he still has Nidhogg Eye 1, this suddenly overwhelms him, and he turns into Nidhogg (or a shade thereof) and flies off, with his story to be concluded in the 3.3 patch.
Which (finally) brings me to my question: how?
Nidhogg's thousand years of hate is performed with Hraesvelgr Eye 1. Nidhogg's eyes are either in the possession of the current Azure Dragoon, or stuck inside a coffin for a thousand years. I would assume this would cause Hraesvelgr Eye 1 to be steeped in hatred, but Hraesvelgr pops it back into its rightful socket without any apparent ill effect.
Since the betrayal of King Thordan a thousand years ago, Nidhogg has not been in possession of his own eyes. He can presumably influence them in some spooky manner from a distance, since Estinien falls prey to it in the level 50 Dragoon Job Quest, and the whole 30-50 Dragoon Job Questline has to do with Estinien carrying Nidhogg Eye 1 away from Ishgard to draw Nidhogg away. Which, in turn, raises the question of why Nidhogg Eye 1 is a higher priority to Nidhogg than Nidhogg Eye 2, resting in the heart of Ishgard.
One possible explanation I can think of is Nidhogg's pain and rage at the point of being de-eyed was potent enough to taint his eyes permanently, and this is reinforced from a distance over a thousand years through dragon magic or some such. Despite this, Hraesvelgr Eye 1, which Nidhogg is actually in possession of, is not as heavily affected, and the WoL's party suffers no ill effects from delivering it back to Hraesvelgr.
Am I missing something? It's been a while since I did the Heavensward 3.0 MSQ, so I don't know if there's an explanation.
Side question: Thordan and his knights (or possibly just some of them) ate Ratatoskr's eyes. What benefit did this give them? If it's mentioned in the game, I probably forgot.