Cell-free Massive MIMO (multiple-input multiple-output) is a promising distributed network architecture for 5G-and-beyond systems. It guarantees ubiquitous coverage at high spectral efficiency (SE) by leveraging signal co-processing at multiple access points (APs), aggressive spatial user multiplexing and extraordinary macro-diversity gain. In this study, we propose two distributed precoding schemes, referred to as local partial zero-forcing (PZF) and local protective partial zero-forcing (PPZF), that further improve the spectral efficiency by providing an adaptable trade-off between interference cancelation and boosting of the desired signal, with no additional front-hauling overhead, and implementable by APs with very few antennas. We derive closed-form expressions for the achievable SE under the assumption of independent Rayleigh fading channel, channel estimation error and pilot contamination. PZF and PPZF can substantially outperform maximum ratio transmission and zero-forcing, and their performance is comparable to that achieved by regularized zero-forcing (RZF), which is a benchmark in the downlink. Importantly, these closed-form expressions can be employed to devise optimal (long-term) power control strategies that are also suitable for RZF, whose closed-form expression for the SE is not available.