Peanut butter is considered a non-perishable food, meaning you do not have to refrigerate it, whether it is "natural" style or not.
And, for what it is worth, the term "natural" is not regulated by the FDA. To know what you are getting in that jar of peanut butter, read the ingredients list. The most "natural" peanut butter will list only peanuts and (usually) salt as ingredients.
See 7 Ways with Peanut Butterfor ideas and tips for using all types of peanut butter.