I can usually tell the difference between a distress call and a egg song. Most roosters I've had make a 'Ahhhhhhhhh' noise ranging from barely a whisper to a loud multi rooster 'I hear/see something above get under something now' when something goes overhead.
The distress call of a hen, personally, sounds more urgent than most egg songs. No build up to everyone else making the same noise.
One thing the does scare me is young chicks making a distressed noise. Sometimes something did happen, like an injury, but in my experience that horrible noise is a chick playing chicken football. One chick finds something 'edible' and doesn't want to share so, a game of keep-a-way happens. It's fun to start and watch a game but, it's not fun to hear that noise randomly thinking your about to see a bloody mess. There is a difference in sound between chicken football and an actual injury I suppose, but it still didn't stop me from running to the brooder and making sure the chicks are ok!
Take some time getting attuned to your birds. Spend a few minutes a day with them see if you can pick apart the behaviors and the sounds that go with them. See who's loud and prone to making a big, loud fuss and who isn't. See if you can tell who's making the noise from sound alone, I can't tell the difference between all of my chickens but some do have unique sounds. Roosters are particularly easy to tell apart by crow. Once they have it down it stays relatively the same for the rest of his life in my experience.
I know how draining it is to constantly worry of the safety of your flock. Take extra precautions if you can't tell the difference between a distress call and a egg song. Anything to make yourself feel better about not checking on them at every noise they make.
Is your flock made up of barred rocks? Or some other striped chicken?