Human languages are compositional, combining smaller units of meaning to express more complex ideas. To explain the emergence of compositionality, researchers have appealed to functional pressures from communication. However, languages may merely inherit the component structure found in the environment. We designed a reference game to explicitly disentangle these possibilities; pairs of participants (N = 450) communicated about about sets of shapes that were assembled from component parts. Critically, we manipulated whether shapes that shared the same parts were competitors within each trial or were distributed across different trials. We found that participants successfully developed efficient conventions for referring to the shapes. However, participants who needed to distinguish shapes that shared components within the same context were more likely to develop compositional systems. When shared components appeared in separate contexts, participants favored non-compositional conventions. These results suggest compositional language structure most readily emerges from immediate communicative pressures rather than environmental structure alone.