The suggestion to modify objects' icons to reflect their status is a common and well-established approach. Adding a "person" symbol to an object's existing icon to indicated that it is being shared, is a great approach.
There are other options, too, of course, such as using a chain symbol as used by Sketch to identify layers that are linked into a design from an external shared library:
With regards to what you call "copied" objects, however, users do expect a "copy" to be its own, stand-alone file, as others have already explained. What you're describing sounds more like macOS's concept of an "Alias", which is an icon that simply points to an actual file.
Multiple aliases can point to the same original file. Clicking any of the aliases opens that original, but users can freely rename, move, copy, and delete aliases without affecting the original at all.
If this is what you need to use in your UI, you can take inspiration from Apple's implementation, which adds a little curved arrow icon to aliases:
Whatever icons you choose, make sure you test them with your users to ensure they understand their respective meanings.