Most existing dialogue systems are user-driven, primarily designed to fulfill user requests. However, in many critical real-world scenarios, a conversational agent must proactively extract information to achieve its own objectives rather than merely respond. To address this gap, we introduce Inquisitive Conversational Agents (ICAs) and develop an ICA specifically tailored to U.S. Supreme Court oral arguments. We propose a Dual Hierarchical Reinforcement Learning framework featuring two cooperating RL agents, each with its own policy, to coordinate strategic dialogue management and fine-grained utterance generation. By learning when and how to ask probing questions, the agent emulates judicial questioning patterns and systematically uncovers crucial information to fulfill its legal objectives. Evaluations on a U.S. Supreme Court dataset show that our method outperforms various baselines across multiple metrics. It represents an important first step toward broader high-stakes, domain-specific applications.