Interferometry is an experimental paradigm underlying a range of applications from atomic clocks to gravitational wave detection and cell-membrane dynamics. The fundamental limit on estimating path differences in interferometers is set by the resources available - the volume of space-time, energy, number of particles - through the Cram\'er-Rao bound. Different metrological schemes are judged by the scaling of the estimator's uncertainty in the resources consumed. Quantum probes are known to provide enhanced precision in single-parameter estimation. However, multiparameter quantum interferometry remains largely unexplored. Here we show that estimating one parameter with quantum-limited precision inexorably leads to a reduced precision of the other. Unlike single-parameter estimation, it is impossible to design a strategy saturating the quantum Cramer-Rao bound for loss and phase estimation in a single setup simultaneously. We design optimal quantum states achieving the best possible simultaneous precisions. Our results reveal general features about concurrently estimating hamiltonian and dissipative parameters, and can be applied to sophisticated sensing scenarios such as quantum imaging.