In spoken question answering, the systems are designed to answer questions from contiguous text spans within the related speech transcripts. However, the most natural way that human seek or test their knowledge is via human conversations. Therefore, we propose a new Spoken Conversational Question Answering task (SCQA), aiming at enabling the systems to model complex dialogue flows given the speech documents. In this task, our main objective is to build the system to deal with conversational questions based on the audio recordings, and to explore the plausibility of providing more cues from different modalities with systems in information gathering. To this end, instead of directly adopting automatically generated speech transcripts with highly noisy data, we propose a novel unified data distillation approach, DDNet, which effectively ingests cross-modal information to achieve fine-grained representations of the speech and language modalities. Moreover, we propose a simple and novel mechanism, termed Dual Attention, by encouraging better alignments between audio and text to ease the process of knowledge transfer. To evaluate the capacity of SCQA systems in a dialogue-style interaction, we assemble a Spoken Conversational Question Answering (Spoken-CoQA) dataset with more than 40k question-answer pairs from 4k conversations. The performance of the existing state-of-the-art methods significantly degrade on our dataset, hence demonstrating the necessity of cross-modal information integration. Our experimental results demonstrate that our proposed method achieves superior performance in spoken conversational question answering tasks.