I guess you are referring to shared functions as opposed to instance functions. There might be edge cases, e.g. the instance constructor of the class containing the webmethod must be called in order to make the instance save. In that case, only instance methods of the class can ever be safe. But that would be a detail of the class implementation.
Of course you cannot ever store user data in shared members of the class, because every user would access that same field. But if you - like mentioned in your example - only access services that differentiate between users with a specific mechanism (like HttpContext.Current.Session) you will note be tempted to run into this mistake. But again, this risk is not specific to the use of shared functions, so my answer is "no".