After much processing, I think I have determined what the sound is. It is Rocky the Squirrel delivering his famous message:

"Bullwinkle is a do..."

And we all know who Bullwinkle is.

Actually, I agree with Braveheart, except that I'm not familiar with the squeaky reel, although I should be. This sound is so far down in the noise that it's almost not there. This suggests crosstalk, radio interference --- or perhaps other 911 dispathers, etc. as other people have said. It could possibly be a bad circuit component. It sounds like AGC (automatic gain control) is cutting in and out as the dispatcher says, "Patsy?"

There are also too many high frequencies for it to have been conversation from the phone that Patsy had just been using, unless John or Burke Ramsey sounded like a cricket in late 1996. Given that the typing sounds are so crisp, the mic that is used on the dispatch end sounds as though it is allowing higher frequencies to pass. This suggests that if the sounds are conversation and assuming they are not radio interference (which can sound very scratchy), they must be coming from the dispatch end, as Maikai has posted.

It might as well be Rocky's famous message....