We investigate the coexistence of underlay spectrum sharing in cell-free massive multiple-input multiple-output (MIMO) systems. A primary system with geographically distributed primary access points (P-APs) serves a multitude of primary users (PUs), while a secondary system serves a large number of secondary users (SUs) in the same primary/licensed spectrum by exploiting the underlay spectrum sharing. To mitigate the secondary co-channel interference inflected at PUs, stringent secondary transmit power constraints are defined for the secondary access points (S-APs). A generalized pilots sharing scheme is used to locally estimate the uplink channels at P-APs/S-APs, and thereby, conjugate precoders are adopted to serve PUs/SUs in the same time-frequency resource element. Moreover, the effect of a user-centric AP clustering scheme is investigated by assigning a suitable set of APs to a particular user. The impact of estimated downlink (DL) channels at PUs/SUs via DL pilots beamformed by P-APs/S-APs is investigated. The achievable primary/secondary rates at PUs/SUs are derived for the statistical DL and estimated DL CSI cases. User-fairness for PUs/SUs is achieved by designing efficient transmit power control policies based on a multi-objective optimization problem formulation of joint underlay spectrum sharing and max-min criteria. The proposed orthogonal multiple-access based analytical framework is also extended to facilitate non-orthogonal multiple-access. Our analysis and numerical results manifest that the primary/secondary performance of underlay spectrum sharing can be boosted by virtue of the average reduction of transmit powers/path-losses, uniform coverage/service, and macro-diversity gains, which are inherent to distributed transmissions/receptions of cell-free massive MIMO.