This study analyzes changes in the attention mechanisms of large language models (LLMs)  when used to understand natural conversations between humans (human-human) . We analyze three use cases of LLMs: interactions over web content, code, and mathematical texts. By analyzing attention distance, dispersion, and interdependency across these domains, we highlight the unique challenges posed by conversational data. Notably, conversations require nuanced handling of long-term contextual relationships and exhibit higher complexity through their attention patterns. Our findings reveal that while language models exhibit domain-specific attention behaviors, there is a significant gap in their ability to specialize in human conversations. Through detailed attention entropy analysis and t-SNE visualizations, we demonstrate the need for models trained with a diverse array of high-quality conversational data to enhance understanding and generation of human-like dialogue. This research highlights the importance of domain specialization in language models and suggests pathways for future advancement in modeling human conversational nuances.