We consider the multiple-input multiple-output (MIMO) wiretap channel under a minimum receiver-side power constraint in addition to the usual maximum transmitter-side power constraint. This problem is motivated by energy harvesting communications with wireless energy transfer, where an added goal is to deliver a minimum amount of energy to a receiver in addition to delivering secure data to another receiver. In this paper, we characterize the exact secrecy capacity of the MIMO wiretap channel under transmitter and receiver-side power constraints. We first show that solving this problem is equivalent to solving the secrecy capacity of the wiretap channel under a double-sided correlation matrix constraint on the channel input. We show the converse by extending the channel enhancement technique to our case. We present two achievable schemes that achieve the secrecy capacity: the first achievable scheme uses a Gaussian codebook with a fixed mean, and the second achievable scheme uses artificial noise (or cooperative jamming) together with a Gaussian codebook. The role of the mean or the artificial noise is to enable energy transfer without sacrificing from the secure rate. This is the first instance of a channel model where either the use of a mean signal or the use of channel prefixing via artificial noise is strictly necessary for the MIMO wiretap channel. We then extend our work to consider a maximum receiver-side power constraint. This problem is motivated by cognitive radio applications, where an added goal is to decrease the received signal energy (interference temperature) at a receiver. We further extend our results to: requiring receiver-side power constraints at both receivers; considering secrecy constraints at both receivers to study broadcast channels with confidential messages; and removing the secrecy constraints to study the classical broadcast channel.