We analyze the frequency-dependent current fluctuations induced into a gate near a quantum point contact or a quantum chaotic cavity. We use a current and charge conserving, effective scattering approach in which interactions are treated in random phase approximation. The current fluctuations measured at a nearby gate, coupled capacitively to the conductor, are determined by the screened charge fluctuations of the conductor. Both the equilibrium and the non-equilibrium current noise at the gate can be expressed with the help of resistances which are related to the charge dynamics on the conductor. We evaluate these resistances for a point contact and determine their distributions for an ensemble of chaotic cavities. For a quantum point contact these resistances exhibit pronounced oscillations with the opening of new channels. For a chaotic cavity coupled to one channel point contacts the charge relaxation resistance shows a broad distribution between 1/4 and 1/2 of a resistance quantum. The non-equilibrium resistance exhibits a broad distribution between zero and 1/4 of a resistance quantum.