I have perfect pitch, and I remember working on it as a kid. I didn't know it was a "thing" you could have, it was just a game I played to see if I could remember what things sounded like. For me, it's not just a memory for pitch, although that's part of it, more a memory for a complete sound. In other words, I've been screwed up many times by a new timbre, like my ear had to learn how to catalogue that color. I think it's useful (for sightreading, memorization), it's fun to play with, it makes for great party tricks too! but is not at ALL essential for great musicianship. Having said that, I bet many musicians have a better memory for notes on their own instrument than they think. They test a few, and if they get them wrong, they conclude, "Well, I don't have perfect pitch." But if you tried to sing the tonic of whatever piece you're playing the most right now, just hearing the beginning in your mind first, and checked it every time you walked by the piano for two weeks, you would probably get an interesting result. I was fascinated by the teacher's post above. Completely inspiring!
One more note: I do know people with PP who are disturbed by sounds that are not perfectly tuned. But the ones I know are people who are disturbed by many other things, too. I think it's personality-driven. Me, I like just intonation. Actually, I love it. So a piano should drive me crazy, and it could, if I let it. But a piano sounds like a piano (certainly I prefer well-tuned, but we all do). A train whistle sounds like a train whistle, and if the telephone ring doesn't coordinate with the dishwasher, so be it. It's a philosophical thing, letting things be what they are. It also helps to know about other tunings and the scales of other cultures, because knowing the broader possibilities for sound lets you relax your expectations. Sorry, this probably wasn't that helpful to the original post. I just get very excited about this topic, because I think most people have more potential for it than they think. Cheers.