Maintaining responsiveness while performing complex task reasoning remains a fundamental challenge in spoken dialogue systems. Existing dual-agent approaches separate conversation from reasoning but typically run the two sequentially, beginning reasoning only after the necessary user information is collected, causing substantial planning latency. We propose Ping-Ponder, a dual-agent framework in which two agents operate concurrently through a shared belief state. The conversational agent (Ping) handles real-time spoken interaction while incrementally extracting intent from user utterances and updating the shared belief state. The reasoning agent (Ponder) begins planning and knowledge retrieval as soon as partial intent information becomes available, without waiting for complete slot filling, and incrementally updates the belief state with intermediate results. This allows Ping to continue eliciting missing information while promptly presenting candidate plans generated by Ponder. In a paired single-operator case study, the proposed approach reduces average planning-phase latency by 93% in English (from 7.1s to 0.5s) and 87% in Japanese (from 7.3s to 1.0s) compared with a sequential dual-agent baseline. In a larger replay study on two spoken task-oriented corpora, the same ordering holds under controlled conditions: Ping-Ponder maintains time-to-first-response parity with the blocking baseline while reducing planning-phase latency on planning-triggered turns by 31–32%, with no evidence of degraded reference-based response quality under an LLM judge; human-perceived conversational quality remains to be tested in a user study. These results suggest that concurrent processing through a shared belief state is a promising direction for reconciling responsiveness and reasoning capability in spoken dialogue systems.