Distributed quantum sensing leverages quantum correlations among multiple sensors to enhance the precision of parameter estimation beyond classical limits. Most existing approaches target phase estimation and rely on a shared phase reference between the signal and the probe, yet many relevant scenarios deal with regimes where such a reference is absent, making the estimation of force or field amplitudes the main task. We study this phase-insensitive regime for bosonic sensors that undergo identical displacements with common phases randomly varying between experimental runs. We derive analytical bounds on the achievable precision and show that it is determined by first-order normal correlations between modes in the probe state, constrained by their average excitations. A single nonclassical bosonic state, coherently distributed across the sensors, can generate these correlations and yield a collective enhancement over the standard quantum limit with a gain linear in the total excitation number. Independent probes can match this noiseless sensitivity only by preparing the corresponding nonclassical resource separately in every mode. We identify multimode states with definite joint parity that saturate the corresponding small-displacement limit and can be probed efficiently via local parity measurements already demonstrated or emerging in several quantum platforms. We further analyze loss, heating, dephasing, and phase jitter, finding that the correlated and independent strategies are affected in complementary ways: the former is less sensitive to loss and heating, whereas the latter is more stable against dephasing and intermode phase fluctuations. Our results are relevant to multimode continuous platforms, including trapped-ion, solid-state mechanical, optomechanical, superconducting, and photonic systems.