Conversational question answering aims to provide natural-language answers to users in information-seeking conversations. Existing conversational QA benchmarks compare models with pre-collected human-human conversations, using ground-truth answers provided in conversational history. It remains unclear whether we can rely on this static evaluation for model development and whether current systems can well generalize to real-world human-machine conversations. In this work, we conduct the first large-scale human evaluation of state-of-the-art conversational QA systems, where human evaluators converse with models and judge the correctness of their answers. We find that the distribution of human-machine conversations differs drastically from that of human-human conversations, and there is a disagreement between human and gold-history evaluation in terms of model ranking. We further investigate how to improve automatic evaluations, and propose a question rewriting mechanism based on predicted history, which better correlates with human judgments. Finally, we analyze the impact of various modeling strategies and discuss future directions towards building better conversational question answering systems.