When a product is duplicated, the URL key is copied over too.
This doesn't mean the copied product is not visible, during the URL reindex, a unique key is generated. If you take a look at Catalog -> URL Rewrite Management and filter by the URL key you will see both keys, and the duplicated product having a suffix (like: -1234, a number).
What can be done from my point of view:
- create an extension that will generate the URL key for the copied product based on the name (like Magento does when you create a new product from scratch)
- implement some other custom logic to generate the URL key for the copied product.