Reconfigurable intelligent surfaces (RIS)-assisted massive multiple-input multiple-output (mMIMO) is a promising technology for applications in next-generation networks. However, reflecting-only RIS provides limited coverage compared to a simultaneously transmitting and reflecting RIS (STAR-RIS). Hence, in this paper, we focus on the downlink achievable rate and its optimization of a STAR-RIS-assisted mMIMO system. Contrary to previous works on STAR-RIS, we consider mMIMO, correlated fading, and multiple user equipments (UEs) at both sides of the RIS. In particular, we introduce an estimation approach of the aggregated channel with the main benefit of reduced overhead links instead of estimating the individual channels. {Next, leveraging channel hardening in mMIMO and the use-and-forget bounding technique, we obtain an achievable rate in closed-form that only depends on statistical channel state information (CSI). To optimize the amplitudes and phase shifts of the STAR-RIS, we employ a projected gradient ascent method (PGAM) that simultaneously adjusts the amplitudes and phase shifts for both energy splitting (ES) and mode switching (MS) STAR-RIS operation protocols.} By considering large-scale fading, the proposed optimization can be performed every several coherence intervals, which can significantly reduce overhead. Considering that STAR-RIS has twice the number of controllable parameters compared to conventional reflecting-only RIS, this accomplishment offers substantial practical benefits. Simulations are carried out to verify the analytical results, reveal the interplay of the achievable rate with fundamental parameters, and show the superiority of STAR-RIS regarding its achievable rate compared to its reflecting-only counterpart.