Camera and motion
The camera takes the shots for your 360° photo, and on devices that have one the LiDAR camera measures how far away things are, so the shots line up. The motion sensors let you look around a photo by turning your device.
The shots are stitched on your device. They are not uploaded anywhere to be processed.
Location
If you allow it, the place where you took a photo and the direction you faced are saved in the photo itself, so you can see where each one was taken.
If you don't allow it, the app works just the same and your photos carry no location. Keep in mind that a photo you share with its location in it tells whoever gets it where it was taken.
Your photos
Your 360° photos are stored on your device. The developer has no access to them.
From version 1.1 on, if your device is signed in to iCloud, your photos are also kept in your own iCloud account, so your gallery is the same on all your devices. Each photo goes there as it is, with its depth and any location saved in it. That storage is private to your Apple Account: Apple runs it, and the developer cannot see what is in it. A photo you delete on one device is deleted on the others too. To stop syncing, switch 360 Image off in your device's iCloud settings; the photos on the device stay.
Saving to Photos asks only to add to your library, never to read it. A photo you open from Photos or Files is the one you pick, and nothing else in them is looked at.
Usage statistics
From version 1.8 on, the app asks whether it may collect usage statistics with Google Analytics. Nothing is collected unless you say yes, and you can change your answer at any time in the Settings app.
The statistics are anonymous: they show which features are used, along with your device model and iOS version, but not who you are. They never include your photos, their locations or anything you type. They are used only to improve the app, never for advertising.
Nothing else
No advertising, no tracking across other apps or websites, no account to create.
Deleting
Deleting a photo in the app removes it from the app. Copies you have saved to Photos or shared stay where you put them. Deleting the app removes everything it stored on the device. Photos synced to iCloud stay in your iCloud account until you delete them there, in your device's iCloud storage settings.