Patient-provider communication has been crucial to cancer patients' survival after their cancer treatments. However, the research community and patients themselves often overlook the communication challenges after cancer treatments as they are overshadowed by the severity of the patient's illness and the variety and rarity of the cancer disease itself. Meanwhile, the recent technical advances in AI, especially in Large Language Models (LLMs) with versatile natural language interpretation and generation ability, demonstrate great potential to support communication in complex real-world medical situations. By interviewing six healthcare providers and eight cancer patients, our goal is to explore the providers' and patients' communication barriers in the post-cancer treatment recovery period, their expectations for future communication technologies, and the potential of AI technologies in this context. Our findings reveal several challenges in current patient-provider communication, including the knowledge and timing gaps between cancer patients and providers, their collaboration obstacles, and resource limitations. Moreover, based on providers' and patients' needs and expectations, we summarize a set of design implications for intelligent communication systems, especially with the power of LLMs. Our work sheds light on the design of future AI-powered systems for patient-provider communication under high-stake and high-uncertainty situations.