There are different chicken vocalizations that mean different things. Mine sometimes give a call very close to the "egg song" when they get nervous or excited, nothing to do with laying an egg or being separated. I think I can tell a difference in the calls sometimes but they really sound about identical. My rooster will sometimes join in too when they get going. Maybe the difference I hear is just the difference in which chicken is singing it, I don't know.
I tend to agree with OGM that it can be a separation call, where are you to the flock. But something I've noticed with a few roosters, not most but a few. When a hen lays the egg and starts the call, the rooster leaves the flock and goes to the hen, he then mates with her, and next leads her back to the flock. This makes sense as a hen that is laying eggs gets fertilized.