We study the pricing of a continuously collateralized credit default swap (CDS). We make use of the "survival measure" to derive the pricing formula in a straightforward way. As a result, we find that, even under a perfect collateralization, there exists an unremovable trace of the counterparty and the investor in the pricing of CDSs due to their default dependence, even though the hazard rates of the two parties are totally absent from the pricing formula. As an important implication, we also study the situation in which the investor enters an offsetting back-to-back trade with another counterparty. We provide simple numerical examples to demonstrate the change in a fair CDS premium level according to the strength of default dependence among the relevant names. We then discuss possible implications for the risk management of the central counterparties.