According to the "Help & Training" documentation (located here), the behavior is supposed to be as follows:
The Clone button on a product quickly creates a new product with the
same information as the existing product. Cloning products also
creates any price book entries in price books that you can view. Price
book entries in price books that you do not have sharing access to
will not be created. (Emphasis mine)
However, I performed a quick test, and it appears that this may not always be true. I created a custom profile that had the following permissions:
Products Tab [Default On]
Chatter Internal User
View Help Link
Allow View Knowledge
Product Read, Create
All other permissions were disabled, including any administrative permissions, etc. Next, I created a user that has no role assigned to it, so it cannot receive sharing by hierarchy. After that, I set Price Book access to Private, so the user has no ability to look at any Price Book not shared to them. I also verified that they could not see any price books at all; they can't even see the tab.
I then logged in as that user and performed a product clone using the standard button. The product page appeared correctly, as no price book entries were displayed on the related list. However, going back to the record as the administrator, I could see that the product did indeed clone all of the line items, contrary to the Help topic I quoted earlier.
So, the correct answer (today) appears to be "prices in price books the user does not have access to will be created," while the correct answer according to the documentation is "prices in price books the user does not have access to will not be created."
I'm going to write support and find out what's going on here; this is one of those times where the observed system behavior is directly opposing the documented behavior of the system.