It depends how well you have written your app. I have not written a subscription application in a while, I use to have a couple of customers that had subscription services available for support, but we have moved them to a different business model.
The last time I was working on these when the user cancels the subscription the API sends back a cancellation request to the website. You will need to grab that request and update your user database that the user is no longer active. When I was testing these, and again it was a couple of years ago the last one that I did, I was surprised at how quick the API from PayPal sent out the cancellation request.
It was being processed and submitted back to the site in less than a couple of seconds. When you think of all the different hooks that are being detached with a subscription cancellation, I was impressed.
So to answer your question, your code in the application will be responsible for processing the cancellation request from the API and updating your user database. It only makes sense since you would not want anyone else having access to your member / customer database.
