Suppose we construct a properly symmetrized multi-particle wave function or operator field in QM or QFT. In the QFT, let's suppose these particles are weakly interacting such that the field is well-described by Fock states.
While the associated particles are technically indistinguishable, if their interaction is weak enough, I'm able to experimentally observe 1 or the other. For example, to take an example from Griffiths, if I have 2 scientists observing electrons in their labs, 1 in Chicago and the other in Los Angeles, we can think of the wave functions and particles as practically distinguishable and hit them with our operators as though the wave-function was not anti-symmetrized, allowing the scientists to observe 'their own' electrons in their experimental systems without worrying about what happens to the other electron.
Where in the operator formalism do we encode this concept of local observation? If I have an N-particle system, where the particles are practically non-interacting, does the vanilla momentum operator 'effectively' tell me the momentum of a single particle, or of both particles, and how does the mathematical formalism account for the fact that while both the particles in Los Angeles and Chicago share the same anti-symmetrized wave function, I can consistently observe different quantities in 1 city and not the other?
It would seem like, if both scientists share the same operators and wave functions, when they observe their electrons, they should always observe the same things, but somehow they've managed to localize at least one of these constructs such that this is definitively not the case.