The short answer is no. In some cases a design studio will want to see education credentials on one's resume; however, there are many, many graphic designers without formal education. So, in general it's not a requirement unless made so by a particular design studio.
There can be definite benefits to design school. Including networking, tools and techniques, principles of design, a historical perspective, exposure to design theory, workflow, etc, etc....
In my own case I had no formal design education, but, I was always artistically inclined and in 1992 landed a production artist opportunity in a video production studio. After a year or so, I started getting overflow design projects as the principal designer. By 1994 my job title was officially "graphic designer". In those days, I was definitely lacking some skills and so took many classes and workshops to fill in the gaps.
Currently I work with an interactive media company as a project director. I'm now rarely involved in "hands-on" design. However, I hire both staff and freelance graphic designers. A graphic design degree can help, but I mainly look at a designer's portfolio and their ability to work on a team. My goal is to hire the right person, and over the years I've hired graphic designers with and without formal design education.
Just to add... When I hire at an entry level, a design degree is a big plus. This is in large part because achieving a design degree means learning to work under deadlines through good workflow and processes. So, at the entry level, without a degree, a candidate must demonstrate those abilities in some other way.