Generally they will not. Some high schools when they send transcripts include a list of what activities you participated in. But usually, they take your word for it.
If they ever find out that you didn't actually do an EC or lied about an award or any from of dishonesty, they will reject you and sometimes notify other schools. And if you are already at the school, and they find out a year a later, you will be kicked out.
But to answer your questions, your exchange school shouldn't worry about it