In human conversation, it is necessary to be brief, while still effectively communicating an idea (Grice). This consequently leads to conversations that leave out a great deal of information, for which each participant will take for granted that the other participant is aware. If this context could not be assumed by the participants, conversation would resemble Tolkien's Ents, where the name for an object encompasses that object's complete history. While this may work in a fictional world, it would be impossible for humans, because even a simple conversation would last far beyond the relevancy of the ideas being communicated. This could lead us to ask how we determine what information may be left out of the conversation. However, considering that the quantity of information included in any conversation is vastly dwarfed by the context omitted, it seems that it might be better to start from nothing, and ask what information should be included.
Let us describe a subset of human knowledge as a collection of experiences and the relationships between them. It seems that conversation is an effort to share that collection of experiences and relationships with the other parties of the conversation. For example, take a simple conversation where John says, "I went to Dairy Queen," and Fred replies, "Oh, what flavor did you get?" Some of the experiences they share include the nature of food, specifically ice cream, and of shops that sell it, and that you can go to them, and so on. In John's collection of experiences, there is a link between one particular visit to Dairy Queen and a particular flavor of ice cream. Fred's collection of experience before the conversation does not include a link between John and the visit in question. John's utterance serves to provide Fred with that connection, so that following it, they share the knowledge that John visited Dairy Queen. Because of Fred's own experiences, he realizes that there is an opportunity for a connection between that visit and the flavor of ice cream had, and for whatever reason, asks a question for which he expects the answer to provide that additional connection.
In fact, it seems that most conversations are of a give-and-take nature that comprise a negotiation to establish connections between shared experiences. If John had instead said, "I went to Tom's place," and Fred is unaware of the fact that Tom's Place is a restaurant, he is more likely to ask, "Who is Tom?" rather than "What did you get?" It is necessary to establish that shared context, before additional connections may be made. In this example, the name "Tom's Place" isn't bound to any specific idea for Fred, and so the connection between John and that place holds little value. John's explanation of the restaurant serves to build those connections in Fred's mind. Following a successful conversation, both parties should share a similar collection of information about experiences and connections between them for the subjects of the conversation.
In that light, it seems that the missing information in conversation is not left out, but rather that the included information is chosen specifically to establish connections that previously did not exist and that the conversation itself consists of a negotiation to ensure that the context is sufficiently shared.
References
Grice, H.P. "Logic and Conversation." 1968 Harvard University Press.