How are you trying to delete or modify the Share (or it’s files/folders) permissions? Using what program on the computer?
Generally one cannot remove the default “Public” Share on the My Cloud nor make it private. Unofficially there are ways to do so. One can use the forum search feature, magnifying glass icon upper right, to find those past discussions…
File and folder permissions are inherited from the top Share the files or folders reside in. Generally one uses the My Cloud Dashboard to configure a Share (other than the default Public Share) to Public or Private. When a Share is set to Private one can assign user permissions to user accounts created through the My Cloud Dashboard. One also uses the My Cloud Dashboard to remove and create top level Shares
If one hasn’t done so it is also a good idea to read the My Cloud User Manual (https://support.wdc.com/product.aspx?ID=904) to learn more about how to configure/use the various features/options of the My Cloud device.