  In closed generic many-body systems, unitary evolution disperses local
quantum information into highly non-local objects, resulting in thermalization.
Such a process is called information scrambling, whose swiftness is quantified
by the operator size growth. However, for quantum systems embedded in an
environment, how the couplings to the environment affect the process of
information scrambling quests revelation. Here we predict a dynamical
transition in quantum systems with all-to-all interactions accompanied by an
environment, which separates two phases. In the dissipative phase, information
scrambling halts as the operator size decays with time, while in the scrambling
phase, dispersion of information persists and the operator size grows and
saturates to an $O(N)$ value in the long-time limit with $N$ the number of
degrees of freedom of the systems. The transition is driven by the competition
between the system intrinsic and environment propelled scramblings and the
environment induced dissipation. Our prediction is derived from a general
argument based on epidemiological models and demonstrated analytically via
solvable Brownian SYK models. We provide further evidence which suggests that
the transition is generic to quantum chaotic systems when coupled to an
environment. Our study sheds light on the fundamental behavior of quantum
systems in the presence of an environment.
