No. You are not required to share your interview. After you finish recording, you'll be given the option to "publish" your interview or to save the interview to your device.
If you choose to publish, the interview and photo will be uploaded to archive.storycorps.org, and they will become publicly accessible and archived by the Library of Congress.
If you choose to save the interview locally, it will only be saved to your device, and it will not be available to the public. We recommend that you back up your interview audio file by exporting it from your device to a desktop or laptop computer (please see instructions here).