We analyze a cavity optomechanical setup, in which the position of an oscillator modulates the internal optical loss. We show that, in contrast to systems with a fixed internal loss, in such a setup, quantum-limited position measurements can be performed and formulate conditions under which it is possible. Additionally, under these conditions the setup exhibits a number of potential benefits for practical operation, including the complete absence of dynamical backaction and optomechanical instability, rejection of classical laser noise, and thermal fluctuations of cavity frequency from the measurement record. We address a few potential experimental implementations of this setup.