Re: Side channel attacks
Traffic analysis isn't usually described as a side channel in the IT security industry, though of course it is one, broadly speaking.
There's a difference between what we might call "first-order" side channels that leak information directly from cryptographic operations, and traffic analysis and other "second-order" channels that leak information about what the entity using the protected data is doing. That difference is largely conceptual, but it does affect what aspects of the system can be compromised.
Traffic analysis, which dates back to antiquity in some forms and is practiced informally in ordinary human interactions, is also much older than side-channel attacks on computation and other machine processes.
Your observation that side-channel information leakage and efficiency are opposed is broadly correct, and arises as a consequence of information thermodynamics. There are some areas where the two can be increased together, such as reversible computing, which can1 reduce both power consumption and the detectable loss of information to the system's environment. On the whole, though, the ultimate defense against side channels is whitening, which means leaking more information, and thus power, than is necessary simply to perform the computation.
For traffic analysis, whitening generally involves generating extra traffic (see e.g. Rivest's "chaffing and winnowing" protocol, which was originally developed to bypass restrictions on encryption but could also be deployed for this purpose).
1In theory, and according to some experiments in practice, though the effect is not large enough to be economically interesting at this time.