In modern times, when we observe nonlocal effects between objects (such as with gravity), we first measure the effect, and only later do we invent something called a "field" whose definition consists of nothing other than the observed measurements themselves.
This is still nothing other than the same "Hypothesis non fingo" as before, except that now, the question of whether fields can possibly be explained in terms of classical physical pictures does not exist anywhere in the entire physics profession. That is, those kinds of hand-waving, natural language questions just don't have any place in the discourse of those who can rightfully be called credentialed, working physicists.
To such physicists, a particle is simply a set of quantum numbers that exists in n-dimensional Hilbert space (or "mathspace"). Again, whatever field is said to exist at the particle's location is nothing more than the measurement that is recorded by some apparatus that exists in [classical] spacetime. So, it is of course necessarily true that the particle and the [part of the] field [that affected it] must exist at the same location.
But to then say that this is therefore a meaningful account of something called "local physics" is, IMO, fairly absurd, because the entire point of the inventions called fields is to say: "Hey, I have no idea what accounts for instantaneous, space-bridging forces like gravity, so I am just going to leave it to the philosophers to offer up speculations so that the masses may be entertained."
Of course, this can be countered by the argument that gravity is fundamentally different from the electromagnetic force, because electromagnetism necessarily depends on the supposed propagation of things called photons. But this argument depends crucially on the notion that our mathematical shorthand for the quantization of the EM field (ie, photons) is at the same time a meaningful physical description that exists in the framework of a classical spacetime picture that is based fundamentally on the quality of locality.
However, QED (quantum electrodynamics; the quantum description of electromagnetism) in no way exists in any such classical framework. It is all rather the very same "mathspace" previously mentioned. The "field" is nothing but whatever bit of data that some classical measuring device recorded there.
p.s. Thanks for the upvotes :)... sometimes it pays to complain, I guess!!!