This paper investigates the performance of downlink simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS)-assisted cell-free (CF) massive multiple-input multiple-output (mMIMO) systems, where user equipments (UEs) are located on both sides of the RIS. We account for correlated Rayleigh fading and multiple antennas per access point (AP), while the maximum ratio (MR) beamforming is applied for the design of the active beamforming in terms of instantaneous channel state information (CSI). Firstly, we rely on an aggregated channel estimation approach that reduces the overhead required for channel estimation while providing sufficient information for data processing. We obtain the normalized mean square error (NMSE) of the channel estimate per AP, and design the passive beamforming (PB) of the surface based on the long-time statistical CSI. Next, we derive the received signal in the asymptotic regime of numbers of APs and surface elements. Then, we obtain a closed-form expression of the downlink achievable rate for arbitrary numbers of APs and STAR-RIS elements under statistical CSI. Finally, based on the derived expressions, the numerical results show the feasibility and the advantages of deploying a STAR-RIS into conventional CF mMIMO systems. In particular, we theoretically analyze the properties of STAR-RIS-assisted CF mMIMO systems and reveal explicit insights in terms of the impact of channel correlation, the number of surface elements, and the pilot contamination on the achievable rate.