Conversational AI is constrained in many real-world settings where only one side of a dialogue can be recorded, such as telemedicine, call centers, and smart glasses. We formalize this as the one-sided conversation problem (1SC): inferring and learning from one side of a conversation. We study two tasks: (1) reconstructing the missing speaker's turns for real-time use cases, and (2) generating summaries from one-sided transcripts. Evaluating prompting and finetuned models on MultiWOZ, DailyDialog, and Candor with both human A/B testing and LLM-as-a-judge metrics, we find that access to one future turn and information about utterance length improves reconstruction, placeholder prompting helps to mitigate hallucination, and while large models generate promising reconstructions with prompting, smaller models require finetuning. Further, high-quality summaries can be generated without reconstructing missing turns. We present 1SC as a novel challenge and report promising results that mark a step toward privacy-aware conversational AI.