I am having a real issue with hunting this bird down, and i'm actually beginning to think my game may have possible glitched.
I found the bird more-or-less straight away but then it flew away. I've hunted high and low for it, day and night and I cannot find it anywhere. I've looked in all the places that have been mentioned in other answers but still got nothing.
Has this happened to anybody else? Does it go anywhere in Soviet Instillation or just around the main 'mill' area?
Best Answer
If you use Survival Instincts the green concentric circles around the messenger bird will appear. If you do not see these circles, even after fast traveling away and then back to the Soviet Installation, then you probably experienced a bug that was fixed in Patch 3.
Patch 3 was released on January 22, 2016. You can read the patch notes here, specifically
I experienced the same issue as you and after patching on January 22, the bird appeared.