As the novel coronavirus spread globally, a growing public panic was expressed over the internet. We examine the public discussion concerning COVID-19 on Twitter. We use a dataset of 67 million tweets from 12 million users collected between January 29, 2020 and March 4, 2020. We categorize users based on their home countries, social identities, and political orientation. We find that news media, government officials, and individual news reporters posted a majority of influential tweets, while the most influential ones are still written by regular users. Tweets mentioning "fake news" URLs and disinformation story-lines are also more likely to be spread by regular users. Unlike real news and normal tweets, tweets containing URLs pointing to "fake news" sites are most likely to be retweeted within the source country and so are less likely to spread internationally.