Compression is essential to storing and transmitting medical videos, but the effect of compression on downstream medical tasks is often ignored. Furthermore, systems in practice rely on standard video codecs, which naively allocate bits between medically relevant frames or parts of frames. In this work, we present an empirical study of some deficiencies of classical codecs on gastroenterology videos, and motivate our ongoing work to train a learned compression model for colonoscopy videos. We show that two of the most common classical codecs, H264 and HEVC, compress medically relevant frames statistically significantly worse than medically nonrelevant ones, and that polyp detector performance degrades rapidly as compression increases. We explain how a learned compressor could allocate bits to important regions and allow detection performance to degrade more gracefully. Many of our proposed techniques generalize to medical video domains beyond gastroenterology.