On a system with paging, a process cannot access memory that it does not own. Why? How could the operating system allow access to other memory?

Because the page is not in its page table The operating system could allow access to other memory by allowing entries for non-process memory to be added to the process page table

Explanation:

On a system with paging, a process cannot access memory that it does not own because the page is not in its page table also the operating system controls the contents of the table, therefore it limits a process of accessing to only the physical pages allocated to the process.

The operating system could allow access to other memory by allowing entries for non-process memory to be added to the process page table. that way two processes that needs to exchange data can efficiently do that. i. e creating a very efficient inter-process communication