In this paper, we rethink the way that communication among users over the Internet, one of the fundamental outcomes of the Internet evolution, takes place. Instead of users communicating directly over the Internet, we explore an architecture that enables users to communicate with (query) Large Language Models (LLMs) that capture the cognition of users on the other end of the communication channel. We present an architecture to achieve such LLM-based communication and we perform a reality check to assess how close we are today to realizing such a communication architecture from a technical point of view. Finally, we discuss several research challenges and identify interesting directions for future research.