While Large Language Models (LLMs) have demonstrated remarkable performance in certain dimensions, their ability to express implicit language cues that human use for effective communication remains unclear. This paper presents ExpressivityArena, a Python library for measuring the implicit communication abilities of LLMs. We provide a comprehensive framework to evaluate expressivity of arbitrary LLMs and explore its practical implications. To this end, we refine the definition and measurements of ``expressivity,'' and use our framework in a set of small experiments. These experiments test LLMs in creative and logical tasks such as poetry, coding, and emotion-based responses. They are then evaluated by an automated grader, through ExpressivityArena, which we verify to be the most pragmatic for testing expressivity. Building on these experiments, we deepen our understanding of the expressivity of LLMs by assessing their ability to remain expressive in conversations. Our findings indicate that LLMs are capable of generating and understanding expressive content, however, with some limitations. These insights will inform the future development and deployment of expressive LLMs. We provide the code for ExpressivityArena alongside our paper.