Open quantum systems can display rich dynamics of quantum information: the information may be scrambled within the system, leaked to an environment, or shared between the system and the environment. In this talk, we discuss the dynamics of quantum information in a generic open system, modeled by local random unitary circuits that are interspersed by projective measurements. The interplay between unitary information scrambling and measurements leads to a sharp phase transition: at sufficiently high rates of measurements, any coherent information in the system is completely lost, while at sufficiently low rates, an extensive amount of information is robustly protected and survives --- this is a consequence of the natural quantum error correction enabled by scrambling dynamics. We will elaborate on how these two phases can be characterized from a variety of complementary perspectives based on the average entanglement entropy within the system, the Fisher information in measurement outcomes, and the quantum channel capacity of the open dynamics. Furthermore, we will present an original method to analyze the dynamics of quantum information in the open system by mapping random unitary circuits with measurements into well-known classical statistical mechanics models. This mapping reveals that the phase transition in a certain limit belongs to the universality class of the bond percolation in the 2D square lattice. We will discuss the implications of our results on characterizing near-term quantum devices.