Empathy is central to human connection, yet people often struggle to express it effectively. In blinded evaluations, large language models (LLMs) generate responses that are often judged more empathic than human-written ones. Yet when a response is attributed to AI, recipients feel less heard and validated than when comparable responses are attributed to a human. To probe and address this gap in empathic communication skill, we built Lend an Ear, an experimental conversation platform in which participants are asked to offer empathic support to an LLM role-playing personal and workplace troubles. From 33,938 messages spanning 2,904 text-based conversations between 968 participants and their LLM conversational partners, we derive a data-driven taxonomy of idiomatic empathic expressions in naturalistic dialogue. Based on a pre-registered randomized experiment, we present evidence that a brief LLM coaching intervention offering personalized feedback on how to effectively communicate empathy significantly boosts alignment of participants' communication patterns with normative empathic communication patterns relative to both a control group and a group that received video-based but non-personalized feedback. Moreover, we find evidence for a silent empathy effect that people feel empathy but systematically fail to express it. Nonetheless, participants reliably identify responses aligned with normative empathic communication criteria as more expressive of empathy. Together, these results advance the scientific understanding of how empathy is expressed and valued and demonstrate a scalable, AI-based intervention for scaffolding and cultivating it.