  We study prepare-and-measure experiments where the sender (Alice) receives
trusted quantum inputs but has an untrusted state-preparation device and the
receiver (Bob) has a fully-untrusted measurement device. A distributed-sampling
task naturally arises in such scenario, whose goal is for Alice and Bob to
reproduce the statistics of his measurements on her quantum inputs using a
fixed communication channel. Their performance at such task can certify quantum
communication (QC), and this is formalised by measurement-device-independent QC
witnesses. Furthermore, we prove that QC can provide an advantage (over
classical communication) for distributed sampling if and only if Bob's
measurements are incompatible. This gives an operational interpretation to the
fundamental notion of measurement incompatibility, and motivates a generalised
notion of it. Our findings have both fundamental and applied implications.
