In natural conversation people sometimes build larger grammatical, semantic and pragmatic units out of multiple turns or installments. The incremental and collaborative character of these `compound contributions' presents challenges for theories of natural language processing. Compounds produced over successive turns by one person have often been analysed in essentially the same way as compounds produced by multiple people. In some recent accounts this putative equivalence has been taken as evidence for the claim that within- and cross-person language processing are fundamentally interchangeable. However, in this paper we present an analysis of compound contributions in a corpus of ordinary dialogues which shows that same- and cross-person compound contributions are constructed in different ways and have different semantic and pragmatic effects on the organisation of dialogue. In particular, we show that they differ in the pragmatic environments in which they occur and that they have different consequences for subsequent turn-taking and interpretation. This asymmetry highlights the need for models of dialogue that account for not just the inherent incrementality of dialogue, but the different status of each contributor towards a turn-in-progress.