On Android, why the images are not getting saved?

You need to have read-write access for writing images to gallery. Make sure you request it with RequestGalleryAccess by passing GalleryAccessMode.ReadWrite mode.

Also, if you are in development mode, there is a chance to have a restricted permission set by unity for write permission. However, this isn't an issue in release mode. In future, we may overcome this issue through our post process scripts.

Do we need to add the permissions in AndroidManifest and info.plist?

No. We automatically add the required permissions on the final builds. All you need to do is just enable the properties as per your usage in Essential Kit Settings.