In this work, we study a full-duplex (FD) cloud radio access network (C-RAN) from the aspects of infrastructure sharing and information secrecy, where the central unit utilizes FD remote radio units (RU)s belonging to the same operator, i.e., the trusted RUs, as well as the RUs belonging to other operators or private owners, i.e., the untrusted RUs. Furthermore, the communication takes place in the presence of untrusted external receivers, i.e., eavesdropper nodes. The communicated uplink (UL) and downlink (DL) waveforms are quantized in order to comply with the limited capacity of the fronthaul links. In order to provide information secrecy, we propose a novel utilization of the quantization noise shaping in the DL, such that it is simultaneously used to comply with the limited capacity of the fronthaul links, as well as to degrade decoding capability of the individual eavesdropper and the untrusted RUs for both the UL and DL communications. In this regard, expressions describing the achievable secrecy rates are obtained. An optimization problem for jointly designing the DL and UL quantization and precoding strategies are then formulated, with the purpose of maximizing the overall system weighted sum secrecy rate. Due to the intractability of the formulated problem, an iterative solution is proposed, following the successive inner approximation and semi-definite relaxation frameworks, with convergence to a stationary point. Numerical evaluations indicate a promising gain of the proposed approaches for providing information secrecy against the untrusted infrastructure nodes and/or external eavesdroppers in the context of FD C-RAN communications.