We propose and analyze continuous measurements of atom number and atomic currents using dispersive probing in an optical cavity. For an atom-number measurement in a closed system, we relate both the detection noise and the heating rate due to measurement backaction to Tan's contact and identify an emergent universal quantum nondemolition (QND) regime in the good-cavity limit. We then show that such a continuous QND measurement of atom number serves as a quantum-limited current transducer in a two-terminal setup. We derive a universal bound on the precision of the current measurement, which results from a trade-off between detection noise and backaction of the atomic current measurement. Our results apply regardless of the strength of interaction or the state of matter and set fundamental bounds on future precision measurements of transport properties in cold-atom quantum simulators.