Deleting photos from one device does not affect the other device unless you are deleting those photos from Photo Stream itself. But having Photo Stream turned on, and then deleting photos from your Camera Roll will not delete any photos from the Photo Stream.
On the other hand, if you are using iCloud Photo Library, anything you delete will be deleted from any other device or Mac using iCloud Photo stream, since you are deleting it from the photo library stored in iCloud which all of those devices are using as well....
So, if that is the case, then you want to turn off iCloud Photo Library to be able to manage your library on a device only.
Cheers,
GB