While not necessarily a prerequisite, many marketing professors have MBA degrees. These professional degrees teach hands-on skills that can be applied to marketing. An MBA program's goal is to teach people how to work in high-level, supervisory positions. So, people with marketing MBAs tend to have the skills required to create and run campaigns. Rather than write copy and compile data, a marketing MBA will be trained in analyzing data, deciding what "big-picture" strategy to take and generally running campaigns.
Many marketing professors have other master's degrees as well. Richard Brookes, for example, a senior lecturer at the University of Auckland in New Zealand, has a Master of Science degree. He researches and writes on branding, marketing strategy and other high-level marketing strategies. His expertise likely comes from a combination of work experience and the analytical skills developed studying for his Master of Science.
The most common degree held by marketing professors is a doctorate. This is a research-heavy, several-years-long postgraduate degree that results in the production of a book-length piece of academic work. This shows strong research skills to a university, both through the fact that the candidate has a Ph.D. and the nature of the research itself. However, a Ph.D. does not necessarily guarantee academic employment; you need to be specialized in an area that the university requires.
A doctorate in marketing itself is not sufficient to teach marketing at a university level. Since doctorates are such specialized degrees, you need to have studied a specific kind of marketing that a university requires research and teaching in. So, for example, the University of Otago's Marketing department's Professor Hoek researched the effect of question wording in surveys for her Ph.D. More broadly, her Ph.D. was in market research. This means that when she was hired, the university needed a professor who had studied market research; had she studied another specialty in marketing, she may not have gotten the job.
So, to teach marketing at a university level you need a doctorate -- but you also need a doctorate in a specialty that a university requires.