The app stores everything solely on your phone. “Everything” isn't much—it’s only the random numbers from your phone’s Bluetooth chirps and the log of those you've gathered from those around you. No one but you has access to this. If you uninstall the app, your proximity log is gone.
You can choose to share a positive test result by inputting a random generated code into the app. This code will be generated by the local public health department managing the app in your area. These random codes aren't meaningful except to the devices which have heard your phone's Bluetooth chirps.
The newly alerted phones will ONLY know that they have recently been in proximity with someone who has tested positive.