Consider a user equipment in a Cell-Free massive Multiple-Input Multiple-Output (CF-mMIMO) system that is served by several Radio Access Points (RAPs). In the uplink of this setup, these RAPs receive noisy observations of the user/source signal and must locally compress their signals before forwarding them to the Central Processing Unit (CPU) through multiple ratelimited fronthaul channels. To retrieve the source signal at CPU, we are interested in maximizing the Mutual Information (MI) between the received signals at CPU and the user/source signal, and purposefully choose the Information Bottleneck (IB)-based compression techniques to design the quantizers at RAPs. We consider both separate and joint designs of the local compressors by establishing basic trade-offs between the informativity and compactness of the outcomes. For the joint design, two different schemes are presented, based on whether to leverage the side-information at CPU. Finally, the effectiveness of both compression schemes will be shown as well by means of numerical investigations over typical digital data transmission scenarios.