CubeTime itself does not collect any data, "personal" or otherwise.
Your solves (data not comprising of personal information including but not limited to the "scramble" used, the date of the solve, the time taken to complete the solve) and your sessions (data including but not limited to named groups of solves, again with no personal data) are stored locally on your device only, unless you are signed into an iCloud account and choose to sync your sessions and solves between iCloud-enabled devices.
Optionally, CubeTime will use Apple iCloud to store your sessions. CubeTime's data that is stored on iCloud can be viewed and deleted in the CubeTime app. This data is handled by Apple's CloudKit service, which is encypted by an account-based key. Your data is stored in a privacy database and we do not have acccess to this data. Additionally, we do not have access to your Apple ID. The Apple iCloud privacy policy is available at https://www.apple.com/legal/privacy/, and further Apple platform security can be viewed here.
The source code of CubeTime is publically viewable at https://github.com/CubeStuffs/CubeTime for independent verification.