The idea behind client server is that the shared responsibility can be shuffled around, for example, we could choose to run certain applications on specific clients and other applications on other clients.
Let us use X11, the X Windows system in Linux, as a typical example of client server flexibility. We have an X11 service running on the server and running on the client we have the client applications such as X Clock or XI.
We could also choose to run XI on one client and not on another client.