The control of a modular multilevel converter can be achieved in various ways. Traditionally, upper-layer energy controllers and middle-layer current controllers are centrally implemented. In some hardware realizations, the submodules also include a local current measurement for purposes of protection and intra-branch energy balancing. In this case, the central controller may also use these measurements for the current control. However, the communication between submodules and the central controller delays will affect the current control's maximum achievable bandwidth. To mitigate this, a distributed implementation is investigated, where the current control is realized locally in the submodule. Analysis of the benefits and drawbacks of this implementation compared to a fully centralized controller are supported by hardware in the loop simulation of a medium voltage test setup.