The Papageno effect concerns how media can play a positive role in preventing and mitigating suicidal ideation and behaviors. With the increasing ubiquity and widespread use of social media, individuals often express and share lived experiences and struggles with mental health. However, there is a gap in our understanding about the existence and effectiveness of the Papageno effect in social media, which we study in this paper. In particular, we adopt a causal-inference framework to examine the impact of exposure to mental health coping stories on individuals on Twitter. We obtain a Twitter dataset with $\sim$2M posts by $\sim$10K individuals. We consider engaging with coping stories as the Treatment intervention, and adopt a stratified propensity score approach to find matched cohorts of Treatment and Control individuals. We measure the psychosocial shifts in affective, behavioral, and cognitive outcomes in longitudinal Twitter data before and after engaging with the coping stories. Our findings reveal that, engaging with coping stories leads to decreased stress and depression, and improved expressive writing, diversity, and interactivity. Our work discusses the practical and platform design implications in supporting mental wellbeing.