Excellent question. To start with the simplest observation, the traditional OS component does remain -- you need a kernel, for example, on both the client and the server.
It is possible, however, to go even further. Consider the Web OS as an interface between the user, the application code, and the user's data. Regardless of where each component of the application runs, where the user is, and where the data resides, the Web OS can tie all of them together.
It is important to realize that both the current PC-centric and the older mainframe-centric way of deploying applications have their advantages. Also, subscribing to a monthly service is the best approach in some cases; in others, it is better to manage software locally. What the free Web OS would offer is the ability to mix-and-match all of these models -- component-by-component, vendor-by-vendor, user-by-user, and situation-by-situation.