Human users increasingly communicate with large language models (LLMs), but LLMs suffer from frequent overconfidence in their output, even when its accuracy is questionable, which undermines their trustworthiness and perceived legitimacy. Therefore, there is a need for language models to signal their confidence in order to reap the benefits of human-machine collaboration and mitigate potential harms. Verbalized uncertainty is the expression of confidence with linguistic means, an approach that integrates perfectly into language-based interfaces. Most recent research in natural language processing (NLP) overlooks the nuances surrounding human uncertainty communication and the biases that influence the communication of and with machines. We argue for anthropomimetic uncertainty, the principle that intuitive and trustworthy uncertainty communication requires a degree of imitation of human linguistic behaviors. We present a thorough overview of the research in human uncertainty communication, survey ongoing research in NLP, and perform additional analyses to demonstrate so-far underexplored biases in verbalized uncertainty. We conclude by pointing out unique factors in human-machine uncertainty and outlining future research directions towards implementing anthropomimetic uncertainty.