Conversational agents promise conversational interaction but fail to deliver. Efforts often emulate functional rules from human speech, without considering key characteristics that conversation must encapsulate. Given its potential in supporting long-term human-agent relationships, it is paramount that HCI focuses efforts on delivering this promise. We aim to understand what people value in conversation and how this should manifest in agents. Findings from a series of semi-structured interviews show people make a clear dichotomy between social and functional roles of conversation, emphasising the long-term dynamics of bond and trust along with the importance of context and relationship stage in the types of conversations they have. People fundamentally questioned the need for bond and common ground in agent communication, shifting to more utilitarian definitions of conversational qualities. Drawing on these findings we discuss key challenges for conversational agent design, most notably the need to redefine the design parameters for conversational agent interaction.