In a standard interferometry experiment, one measures the phase difference between two paths by recombining the two wave packets on a beam-splitter. However, it has been recently recognized that the phase can also be estimated via local measurements, by using an ancillary particle in a known superposition state. In this work, we further analyse these protocols for different types of particles (bosons or fermions, charged or uncharged), with a particular emphasis on the subtleties that arise when the phase is due to the coupling to an abelian gauge field. In that case, we show that the measurable quantities are spacetime loop integrals of the 4-vector potential, enclosed by two identical particles or by a particle-antiparticle pair. Furthermore, we generalize our considerations to scenarios involving an arbitrary number of parties performing local measurements on a general charged fermionic state. Finally, as a concrete application, we analyse a recent proposal by Marletto and Vedral (arXiv:1906.03440).