No. Instagram doesn't let the person who posted the story know if you screenshot it on mobile.
No, they can't. Instagram's mobile app doesn't have a built - in function to inform the user when their story is screenshotted. This is in line with the general privacy settings of the app. While some people might worry that there could be some hidden way to detect screenshots, Instagram has made it clear that for the moment, taking a screenshot of a story is a private action that the story creator won't be informed about. So, feel free to screenshot those interesting Instagram stories on your mobile without the fear of being caught.