Who Really Killed the Witch King in ROTK??

[Gordis]

Originally Posted by Alcuin

But at least some of the Nazgûl must have been in the personal guard, I think.
Sauron’s personal guard must be first peeled away in order to get to him: that’s the purpose of having a personal guard.
Any Nazgûl in Sauron’s personal guard would have been cut down or (once Sauron was vanquished) forced to try to escape literally unseen.

That makes the nazgul's 100% survival even harder to understand. Were the Nazgul literally impossible to kill without proper enspelled weapons? Or did they flee, regardless of Sauron's mind-control. (The Dark Lord likely had forgotten about them at the moment).
Otherwise they would have been cut down with the rest of Sauron's guard. After all, the personal guard of Elendil, Gil-Galad etc. the nazgul were fighting with were also quite tough guys, no doubt. And there may have been other Elf lords and Numenorean lords who remained to fight the nazgul: Celeborn, Thranduil, Amroth, Meneldur...

Originally Posted by DPR

When Sauron was thrown down, how do you suppose the Nazgul felt, lost or free?

I think that even before Sauron had been cut down, there had to be a moment when he forgot about the nazgul, minding only his own hide. The nazgul may have felt disoriented, as the orcs felt at the Black Gate in TA 3019, when Sauron's will left them, wholly concentrating on Frodo at Sammath Naur. There might have been a moment when the nazgul asked themselves: "What in Ungoliant are we doing here fighting these Elves?"
Perhaps it was at that moment that the nazgul "passed into shadows" - i.e. used their Rings to escape. Later they had enough time to assess the new situation and start to enjoy freedom.
I agree with NEL's answer, with a slight correction:

Originally Posted by Alcuin

The text indicates that Boromir I, Steward of Minas Tirith and ancestor of Boromir in LotR, was not afraid of them either, and Faramir collapsed only because of Black Breath in connection with physical and emotional exhaustion in combination with an arrow wound.)

I think you don't make enough distinction between not being afraid and having enough guts to suppress one's fear and not act upon it. I think with the guys you have mentioned it was always the latter. The lucky Elves might have indeed felt no fear, but humans always did - it was the very essence of nazgul, former Men, that caused this horror in humans.

But suppressing one's horror again and again is unhealthy, maybe even more draining than surrendering to it. It is exhausting, leads to neurosis and what in ME is called "effects of the Black Breath".