Based on the research presented here: "Hover and wind-tunnel testing of shrouded rotors for improved micro air vehicle design" and other reading that translates similarly well-designed shroud and rotor combinations into more general layman's terms, I have found that some authors indicate that there is a performance increase of 40% to 60% in static thrust for such a configuration. One source has indicated a theoretical increase in static thrust as much as 80%.
If we focus our attention on a multirotor design that is intended as a craft whose principal purpose is for inspections of stationary objects such as buildings and stationary infrastructure, and it is not expected to fly fast nor fly far, then the primary benefit from the shrouded rotor is the improvement of static thrust. Flight in translation will be rare and incidental to maneuvering around the object being inspected.
And assuming that the shroud can be manufactured with a mass that is less that the incremental improvement in thrust, this design will serve its intended purpose with the significant benefit of having some built-in protection from horizontal impact with obstacles.
The need for maneuverability will be equal to most other multirotor applications intended for inspections and as such yaw will still need to be produced by controlling the speed of the propellers by the flight controller.
What impact will the shroud have on the production of yaw by the torque produced by the propellers (rotors)?