Open-domain dialogue systems have made promising progress in recent years. While the state-of-the-art dialogue agents are built upon large-scale social media data and large pre-trained models, there is no guarantee these agents could also perform well in fast-growing scenarios, such as live streaming, due to the bounded transferability of pre-trained models and biased distributions of public datasets from Reddit and Weibo, etc. To improve the essential capability of responding and establish a benchmark in the live open-domain scenario, we introduce the LiveChat dataset, composed of 1.33 million real-life Chinese dialogues with almost 3800 average sessions across 351 personas and fine-grained profiles for each persona. LiveChat is automatically constructed by processing numerous live videos on the Internet and naturally falls within the scope of multi-party conversations, where the issues of Who says What to Whom should be considered. Therefore, we target two critical tasks of response modeling and addressee recognition and propose retrieval-based baselines grounded on advanced techniques. Experimental results have validated the positive effects of leveraging persona profiles and larger average sessions per persona. In addition, we also benchmark the transferability of advanced generation-based models on LiveChat and pose some future directions for current challenges.