Data minimisation is a privacy enhancing principle, stating that personal data collected should be no more than necessary for the specific purpose consented by the user. Checking that a program satisfies the data minimisation principle is not easy, even for the simple case when considering deterministic programs-as-functions. In this paper we prove (im)possibility results concerning runtime monitoring of (non-)minimality for deterministic programs both when the program has one input source (monolithic) and for the more general case when inputs come from independent sources (distributed case). We propose monitoring mechanisms where a monitor observes the inputs and the outputs of a program, to detect violation of data minimisation policies. We show that monitorability of (non) minimality is decidable only for specific cases, and detection of satisfaction of different notions of minimality in undecidable in general. That said, we show that under certain conditions monitorability is decidable and we provide an algorithm and a bound to check such properties in a pre-deployment controlled environment, also being able to compute a minimiser for the given program. Finally, we provide a proof-of-concept implementation for both offline and online monitoring and apply that to some case studies.