I have never come across a "Viral Marketing Specialist" role or anything like that, but it's a skill that lots of marketers are required to have nowadays, especially those working in agencies and startups. There's a new marketing function today called "Growth Hacker", which is the person in a startup whose full north is user growth. That person will surely use loads of viral marketing techniques to do their job. But it's always worth mentioning that the basic marketing skills, like understanding consumer behaviour and measuring results are a still huge part of anyone trying to insert virality in any idea.