Data visualizations play a critical role in both communicating scientific evidence about climate change and in stimulating engagement and action. To investigate how visualizations can be better utilized to communicate the complexities of climate change to different audiences, we conducted interviews with 17 experts in the fields of climate change, data visualization, and science communication, as well as with 12 laypersons. Besides questions about climate change communication and various aspects of data visualizations, we also asked participants to share what they think is the main takeaway message for two exemplary climate change data visualizations. Through a thematic analysis, we observe differences regarding the included contents, the length and abstraction of messages, and the sensemaking process between and among the participant groups. On average, experts formulated shorter and more abstract messages, often referring to higher-level conclusions rather than specific details. We use our findings to reflect on design decisions for creating more effective visualizations, particularly in news media sources geared toward lay audiences. We hereby discuss the adaption of contents according to the needs of the audience, the trade-off between simplification and accuracy, as well as techniques to make a visualization attractive.