Should they? I don’t think so. However, I think they should understand that there are certain principles of web design that do not apply to other disciplines of design, and vice versa.

The argument isn’t black and white though, I’m seeing a lot of arguments against Frank Chimero like this:

To be a print designer, you don’t need to understand how to use a letterpress.

Well no, of course you don’t. But you need to understand that, like anything, that letterpress has certain limitations and abilities. It CAN do A & B but it most certainly can’t do X & Y; as a print designer, you should know that.

In a nutshell

I know a lot of web designers who can code and a lot who don’t want to (or can’t). All of the talented ones understand the core principles of web design such as the box model and that the web is largely pixel-based rather than vector-based. That’s why they produce great designs.