A three-terminal Josephson junction biased at opposite voltages can sustain a phase-sensitive dc current carrying three-body static phase coherence, known as the “quartet current.” We calculate the zero-frequency current noise cross correlations and answer the question of whether this current is noisy (like a normal current in response to a voltage drop) or noiseless (like an equilibrium supercurrent in response to a phase drop). A quantum dot with a level at energy ε0 is connected to three superconductors Sa,Sb, and Sc with gap Δ, biased at Va=V,Vb=−V, and Vc=0, and with intermediate contact transparencies. At zero temperature, nonlocal quartets (in the sense of four-fermion correlations) are noiseless at subgap voltage in the nonresonant dot regime ε0/Δ≫1, which is demonstrated with a semianalytical perturbative expansion of the cross correlations. Noise reveals the absence of granularity of the superflow splitting from Sc towards (Sa,Sb) in the nonresonant dot regime, in spite of finite voltage. In the resonant dot regime ε0/Δ≲1, cross correlations measured in the (Va,Vb) plane should reveal an “anomaly” in the vicinity of the quartet line Va+Vb=0, related to an additional contribution to the noise, manifesting the phase sensitivity of cross correlations under the appearance of a three-body phase variable. Phase-dependent effective Fano factors Fφ are introduced, defined as the ratio between the amplitudes of phase modulations of the noise and the currents. At low bias, the Fano factors Fφ are of order unity in the resonant dot regime ε0/Δ≲1, and they are vanishingly small in the nonresonant dot regime ε0/Δ≫1.