Agent interaction has long been a key topic in psychology, philosophy, and artificial intelligence, and it is now gaining traction in large language model (LLM) research. This experimental study seeks to lay the groundwork for our understanding of dialogue-based interaction between LLMs: Do persona-prompted LLMs show consistent personality and language use in interaction? We condition GPT-3.5 on asymmetric personality profiles to create a population of LLM agents, administer personality tests and submit the agents to a collaborative writing task. We find different profiles exhibit different degrees of personality consistency and linguistic alignment in interaction.