While beside is a synonym for besides, there are some phrases which use the word and are clearly defined one way or the other.
Beside oneself means to be overwhelmed with emotion, usually sadness, anger, or frustration. The oneself is reflexive, meaning it changes to himself or herself depending on the subject.
The phrase can be used by itself. However, for clarity or simply because of habit, the phrase is usually qualified by the emotion one is distraught by. For example, beside himself with anger or beside herself with anguish.