How to act based on a styled image is generated? three image styles defined: thumb, medium and large. 
By uploading an image, its thumb/medium/large copies are not generated. .    each styled copy of image is generated only when specific style is asked to be viewed. 

I want to do something after styled copy is generated. For example, I want to upload medium size of image to my other server with my API copyToMyServer(). 

How may I do so? Couldn't find hook for this case.
I believe this is what you are looking for.
http://happypixels.com/blog/triggering-creation-imagecache-derivatives-programmatically

Also see the trigger module.
http://drupal.org/documentation/modules/trigger

You can add your desired action to trigged.