No, the BrightSentinel mobile app does not need storage space on your phone to store the data from the AnnoLogs.
The mobile app communicates with the AnnoLog sensors via the Bluetooth (part of your phone and commonly used for other applications - e.g.: earphones) and instructs the AnnoLog to stop, start and spit out its data, which the AnnoLog then flings out to the BrightSentinel Cloud directly for storage, analysis and reporting.
This centralised storage is particularly useful if you are working in one location and your colleagues need to acess the data at another location.