Resonant scattering spectrometers (RSSs) have been used to make high-precision full-disc helioseismic observations since the 1970s. They are capable of very high-precision determinations of line-shift, meaning that they are used to obtain precise velocity measurements, and, for suitably configured RSSs, the disc-averaged longitudinal magnetic field (SMMF). In order to exploit fully the very extensive high-precision data sets, it is essential to understand the noise characteristics of the instruments. This paper re-examines the consequences for velocity and SMMF determinations of there being noise on the scattered light signals measured by an RSS. It presents a theoretical description, and analysis of simulations, which match well the noise variations found recently in both BiSON velocity and SMMF observations. It also outlines a method for using the differences between the power spectra of redundant channels to analyse instrumental noise characteristics.