Abstract: Despite the fact that the Lambda CDM model has been highly successful over the last few decades in providing an accurate fit to a broad range of cosmological and astrophysical observations, different intriguing tensions and anomalies emerged at various statistical levels. Given the fact that the dark energy and the dark matter sectors remain unexplored, the answer to some of the tensions may rely on modifications of these two dark sectors. This manuscript explores the important role of the growth of structure in constraining non-standard cosmologies. In particular, we focus on the interacting dark energy (IDE) scenario, where dark matter and dark energy interact non-gravitationally. We aim to place constraints on the phenomenological parameters of these alternative models, by considering different datasets related to a number of cosmological measurements, to achieve a complementary analysis. A special emphasis is devoted to redshift-space distortion measurements (RSD), whose role in constraining beyond the standard paradigm models has not been recently highlighted. These observations indeed have a strong constraining power, rendering all parameters to their Lambda CDM canonical values, and therefore leaving little room for the IDE models explored here.