The ability to engage in mixed-initiative interaction is one of the core requirements for a conversational search system. How to achieve this is poorly understood. We propose a set of unsupervised metrics, termed ConversationShape, that highlights the role each of the conversation participants plays by comparing the distribution of vocabulary and utterance types. Using ConversationShape as a lens, we take a closer look at several conversational search datasets and compare them with other dialogue datasets to better understand the types of dialogue interaction they represent, either driven by the information seeker or the assistant. We discover that deviations from the ConversationShape of a human-human dialogue of the same type is predictive of the quality of a human-machine dialogue.