In this paper, we first introduce the notion of channel leakage as the minimum mutual information between the channel input and channel output. As its name indicates, channel leakage quantifies the minimum information leakage to the malicious receiver. In a broad sense, it can be viewed as a dual concept of channel capacity, which characterizes the maximum information transmission to the targeted receiver. We obtain explicit formulas of channel leakage for the white Gaussian case, the colored Gaussian case, and the fading case. We then utilize this notion to investigate the fundamental limitations of obfuscation in terms of privacy-distortion tradeoffs (as well as privacy-power tradeoffs) for streaming data; particularly, we derive analytical tradeoff equations for the stationary case, the non-stationary case, and the finite-time case. Our results also indicate explicitly how to design the privacy masks in an optimal way.