Measuring the environments of massive galaxies at high redshift is crucial to understanding galaxy evolution and the conditions that gave rise to the distribution of matter we see in the Universe today. While high-z radio galaxies (HzRGs) and quasars tend to reside in protocluster-like systems, the environments of their radio-quiet counterparts are relatively unexplored, particularly in the submillimetre, which traces dust-obscured star formation. In this study, we search for 850 μm-selected submillimetre galaxies (SMGs) in the environments of massive ($M_{\star }$$\gt 10^{11}$${\rm M}_{\odot }$), radio-quiet ($L_{500 {\rm MHz}}$$\lesssim 10^{25}$ WHz$^{-1}$) galaxies at $z \sim 1\!-\!3$ using data from the SCUBA-2 COSMOS (S2COSMOS) survey. By constructing number counts in circular regions of radius 1-6 arcmin and comparing with blank-field measurements, we find no significant overdensities of SMGs around massive radio-quiet galaxies at any of these scales, despite being sensitive down to overdensities of $\delta \sim 0.4$. To probe deeper than the catalogue we also examine the distribution of peaks in the SCUBA-2 signal-to-noise (SNR) map, which reveals only tentative signs of any difference in the SMG densities of the radio-quiet galaxy environments compared to the blank field, and only on smaller scales (1 arcmin radii, corresponding to $\sim 0.5$ Mpc) and higher SNR thresholds. We conclude that massive, radio-quiet galaxies at cosmic noon are typically in environments with $\delta \lesssim 0.4$, which are either consistent with the blank field or contain only weak overdensities spanning sub-Mpc scales. The contrast between our results and studies of HzRGs with similar stellar masses and redshifts implies an intrinsic link between the wide-field environment and the radio luminosity of the active galactic nucleus at high redshift.