Design is not about style and create beautiful websites (beauty is so subjective). Design is design is not art and is about create functional solutions that are creative, pleasant to the eyes and elegant, is not taste is creativity and creativity requires the understanding and familiarity with the media, tools and techniques.
Should a Designer, code? not necessary; is important to a Designer to know about the technical possibilities and limitations? yes absolutely.
Reverting the question, should a Developer, design? same as in the Designers case.
Designers need to understand technical possibilities and limitations to envision and create the best User Interaction and style for the Site, but as well the Designer requires a good enough knowledge on the field of the site (does it need to be an expert, absolutely not but it can help) that is being created in order to understand what it will be the best User Experience (no is not about looking few websites and and copy bits and pieces, it goes beyond that and that is a long discussion).
Developers needs to understand design for many reasons and one is understand how to realize that idea the most efficient way and sometimes indicate limitations on the platform that could be a risk for the project and provide possible solutions for implementation of that "beautiful" design.
Do help for a Designer know HTML/CSS yes off course but is Ok if is just enough to hack a page to validate design assumptions.
Should someone hire a designer that knows how to code, expecting to save money not paying a Developer, then you could run the risk of having a half baked solution unmaintainable code and take even longer to deliver.
I know Designers that can create amazing code, and be able to learn the entire aspect doing every single piece of code and create amazing sites, as well I know Developers that can come with really elegant functional solutions and designs.
At the end is what makes you feel comfortable and you are willing to learn and gain experience on it.