I'm sure Bungie probably got the Headless screeches off of some stock sound CD as they did so many Marathon sounds, but I recently discovered entirely by chance what those sounds actually are.
There's a type of sea bird in my area called the Snowy Plover, that's apparently an endangered species, but you can still go walking on the beach near their nesting grounds. The only other birds around there are brows pelicans and western gulls, and their calls are uninteresting and can be heard all over. But walking down along that beach the past couple of days, I kept hearing Headless screeches (the kind of sound they make when you shoot them). I traced the source of them to see if maybe someone was playing PiD on the beach, but no such luck... it's just bird calls! And since the only non-gull, non-pelican around there are the Snowy Plovers, it's gotta be them. Being endangered would explain why I've never heard them before I moved here.
I think that the Headless screeches may have received a bit of post-processing on top of the regular Plover calls, but they're so damned similar, the same exact pitch-shift for *both* of the headless screeches, that I'm sure the Headless sounds were taken from these Plovers.