The Flower App collects no information of any sort, and passes none to any external server or any other data management system.
Information you enter into the App's Search Toolbar is saved within the App as a convenience for your later use. Only the most recent entry is saved, not a complete history of those searches. If you are concerned someone might use your device to see those searches you should not enter sensitive information there. As a practical note, the only information searched is the content of the Flower App and it contains no sensitive information.
Similarly, the content of the app's Flower Log is available when your device is connected to a computer. You should not save sensitive information in the Flower Log as no special security measures have been taken to protect it.