Cell-free massive multiple-input multiple-output (MIMO) networks, where a massive number of geographically distributed access points cooperate to jointly serve the users, have been proved to solve many of the interference related issues associated to cellular networks. One of the main challenges of cell-free massive MIMO networks, however, is to guarantee that the potential benefits this architecture entails can be achieved with signal processing computational complexities and fronthaul resource requirements that are scalable as the number of users goes to infinity. User-centric architectures have been proposed in the literature that potentially solve the scalability issues related to, first, the signal processing associated to channel estimation and massive MIMO combining/precoding, second, the power allocation algorithms and, third, the fronthaul signaling for data and channel state information sharing. One of the best centralized combiners/precoders that have been proposed in practice, however, which is termed as partial minimum mean square error (P-MMSE), does not allow obtaining expressions of its spectral efficiency in a scalable way except when using lower bounds based on the well-known use-and-then-forget (UatF) approach. These lower bounds, however, in addition to being rather inaccurate, are not suited for the design and analysis of non-linear combining/precoding schemes based on successive interference cancellation (SIC) when users are equipped with multiple antennas. In this paper, we propose novel linear and non-linear combining/precoding schemes, which we term improved P-MMSE (IP-MMSE), whose achievable spectral efficiency can be accurately analyzed in a scalable manner. The proposed IP-MMSE combiner/precoder achieves spectral efficiencies close to that provided by MMSE-based unscalable solutions, and outperforms to a great extent the P-MMSE-based counterparts.