We investigate multiparameter quantum estimation protocols based on measurement-after-interaction (MAI) strategies, in which the probe state undergoes an additional evolution prior to linear measurements. As we show in our study, this extra evolution enables different level of advantages depending on whether it is implemented locally or nonlocally across the sensing nodes. By benchmarking MAI strategies in both discrete- and continuous-variable systems, we show that they can significantly enhance multiparameter sensitivity and robustness against detection noise, particularly when non-Gaussian probe states are employed, cases where standard linear measurements are often insufficient. We also derive analytical results for multiparameter squeezing and establish the corresponding scaling laws for spin-squeezed states, demonstrating that MAI protocols can reach the Heisenberg scaling. These results pave the way for immediate experimental implementation in platforms such as atomic ensembles and optical fields.