To obviate the control of reflective intelligent surfaces (RISs) and the related control overhead, recent works envisioned autonomous and self-configuring RISs that do not need explicit use of control channels. Instead, these devices, named hybrid RISs (HRISs), are equipped with receiving radio-frequency (RF) chains and can perform sensing operations to act independently and in parallel to the other network entities. A natural problem then emerges: as the HRIS operates concurrently with the communication protocols, how should its operation modes be scheduled in time such that it helps the network while minimizing any undesirable effects? In this paper, we propose an orchestration framework that answers this question revealing an engineering trade-off, called the self-configuring trade-off, that characterizes the applicability of self-configuring HRISs under the consideration of massive multiple-input multiple-output (mMIMO) networks. We evaluate our proposed framework considering two different HRIS hardware architectures, the power- and signal-based HRISs that differ in their hardware complexity. The numerical results show that the self-configuring HRIS can offer significant performance gains when adopting our framework.