We study how preferences for migration-related narratives differ between private and public contexts and how social media fuel opinion polarization. Using a German representative sample (n=1,226), we found that individuals, especially from the left and center, avoided publicly endorsing anti-migration narratives. In an experiment on Twitter (n=19,989) we created four Twitter profiles, each endorsing one of the narratives. Far-right users exhibited markedly different engagement patterns. While initial public endorsements, measured by follow-back rates, aligned with private preferences, social media interactions amplified support for the most hostile and polarizing narrative. We conclude that social media significantly distort private preferences and amplify polarization.