OnionShare doesn't collect any information about its users. The desktop and mobile apps don't collect any analytics and don't do any behavioral tracking. The OnionShare website, onionshare.org, stores two weeks of web server logs, but these logs don't include IP addresses.
Third party app stores, such as the App Store for iPhone, Google Play for Android, and Snapcraft for Linux, do collect information about their users. If you download OnionShare from one of these app stores, refer to their privacy policies.