Screeching sound? Like a high pitched whine or a really loud one?
Which ever the case may be, this sounds like a case of bad coil whine or just normal coil whine depending on which game you are playing and the scenarios this occurs in.
Put on some overlay which displays your fps ingame and see if the framerate is absurdly high once the noise happens. This can be done with softwares such as MSI Afterburners overlay and many others.
Some games start up with unlimited fps which results in framerates such as +2000 which causes terrible coilwhine, but once you pass the menu or sponsor logos the noise stops because the fps goes down alot.
Should also be important to note that coil whine varies from card to card.
Some cards may start producing noise at framerates such as 2000 to 3000 while others may start at just 100 to 300.
But since the noise is only audible when launching the game, i really think it is just coil whine which is nothing dangerous and will not harm your GPU or other components in any way if thats what you are worrying about.
No way for me to know if its coil whine though since there isnt an audio file on your post but by the way you are describing it im 99% sure it is just that.