Let me preface by saying that I love Curse of the Black Pearl more than probably any other movie. I've also seen it way more times than I can count. And I think the movie is basically flawless with the exception of this detail.
There are a number of times where we see contradictory responses of the cursed pirates to pain. In the beginning of the movie, a pirate takes a tomahawk in the back and cries out in pain; Ragetti gets hot coals dumped on him and exclaims "It's hot!" Later in the movie, Will impales a pirate with a massive pole and he cries out in pain as well. In contrast, Barbossa gets impaled twice and shot and hardly reacts. Pintel gets shot and is unphased.
Barbossa, for his part, explains that "I feel nothing. Not the wind on my face, nor the spray of the sea, nor the warmth of a woman's flesh." So what gives?
There are basically four options: 1) The pirates can feel pain but not pleasure. 2) The pirates are faking it for some reason. 3) The pirates don't feel anything, but there is some explanation as to why they appear to feel pain. Or 4) This is just a bona fide plot hole.
It is much more coherent with the nature of the curse that the pirates don't feel pain. The condition of the cursed pirates is a kind of sterilized existence - they are alive insofar as they are sentient, but cannot experience any of the sensations that make life worth living (even though they still have the desire). This includes pain - if they could feel pain, they could at least feel something. They inhabit this subhuman existence: "We are not among the living and so we cannot die, but neither are we dead." I would liken it to anesthetics used by doctors - I went into my dentist to get a tooth pulled, and they gave me laughing gas to numb my mouth. I could still feel that tugging, but it wasn't pain. I think it is a similar experience for other surgeries where they need you awake. That is how the pirates exist - they feel something, but not in the sense of being truly alive (and this is true for every sensation, not just pain).
So why do the pirates appear to experience pain? I can give a plausible explanation. It is a kind of instinctual reaction - for example, when you get burned or drop something on your foot and cry out before the pain even hits you, because you are expecting the pain. The pirates have that expectation of pain, which is why they cry out, but they never actually feel it. Barbossa may just be so used to it that it doesn't phase him, and Pintel was not really expecting that pain since Barbossa's gunshot was so out of the blue.
However, if you asked me: Do I really believe the screenwriters thought this deep into the metaphysics of a magic pirate curse? The answer is no. Which doesn't necessarily mean there is no plausible explanation (see above), but I think the pirates' pain was just a simple detail that was overlooked, either for the sake of plot or out of carelessness. In the invasion of Port Royal, Will tomahawks a pirate who then comes back. This plants the seeds of suspicion in the audience: didn't Will already kill this guy? What's going on? But if the pirate simply takes a tomahawk in the back with no reaction, that gives away the game. Same with the hot coals on Ragetti's head: he returns to pursuing Elizabeth moments later, which will raise some red flags for the audience because any normal person would be incapacitated (although he still shows pain). The pirate screaming after being impaled by Will does not advance the plot - this is just pure carelessness. Jack's expression after being stabbed is more shocked than anything else so I think this is negligible.
It is frustrating because they absolutely could have made the movie without this contradiction at all. When the pirate tells Will "say goodbye!" and then gets obliterated by the falling sign, that could have been the pirate that reappeared to Will; the seeds of suspicion are planted in the audience, without giving away the game. The other times in the movie where cursed pirates apparently experience pain are negligible for the plot and could have been eliminated entirely. But this movie is wonderful for so many other reasons, and I can suspend disbelief over this one quandary to enjoy everything else about it.