I used to call them in often, in the woods, not the open. Shot more than I want to remember, on a regular basis. Shots were always 100 yards or closer. In those days I used old military cartridge rifles.
I have never heard an actual dying rabbit in the woods either...but it sure brings them in. Last cougar I got came in to a dying cottontail call, right after I changed to a dying jack rabbit call. ? Had another come in to the same thing, cotton tail for a while, then switch to Jack. He came within ten-15 feet...and got away! For sure, if I was out in the woods, and heard a dying rabbit, I'd be 99.9999999% sure it was a hunter. But I have called in two hunters, who had no idea what the noise was, and came to investigate. I shut off the call, and remained hidden, and got a good laugh at their bewilderment.
I've never heard a "fawn in distress" in the woods, but that call brought two wolves within 20 feet of me. !!