To promote engagement, recommendation algorithms on platforms like YouTube increasingly personalize users' feeds, limiting users' exposure to diverse content and depriving them of opportunities to reflect on their interests compared to others'. In this work, we investigate how exchanging recommendations with strangers can help users discover new content and reflect. We tested this idea by developing OtherTube -- a browser extension for YouTube that displays strangers' personalized YouTube recommendations. OtherTube allows users to (i) create an anonymized profile for social comparison, (ii) share their recommended videos with others, and (iii) browse strangers' YouTube recommendations. We conducted a 10-day-long user study (n=41) followed by a post-study interview (n=11). Our results reveal that users discovered and developed new interests from seeing OtherTube recommendations. We identified user and content characteristics that affect interaction and engagement with exchanged recommendations; for example, younger users interacted more with OtherTube, while the perceived irrelevance of some content discouraged users from watching certain videos. Users reflected on their interests as well as others', recognizing similarities and differences. Our work shows promise for designs leveraging the exchange of personalized recommendations with strangers.