The idea of a field can be conceived of in multiple ways. I will describe the one that I prefer.
I need to point out first: that which can be actually measured is motion. What we observe is that under particular circumstances the motion of objects is affected (acceleration). We can attribute that affect to a field, but that field cannot be measured directly. In that sense we cannot assert the existence of the field with the same certainty as we have for the observations of actual motion.
The concept of a field is used because it affords an economy of thought. Throughout the history of physics it has often proved to be a good policy to proceed in a direction that affords economy of thought,
The concept of a field is not as if a particle casts a net out towards the universe.
Rather, the field is conceived as an entity that exists anyway, but in the absence of an interaction source it is in a uniform state.
In the case of the electrostatic interaction: presence of electric charge causes a distortion from uniform state in the space adjacent to that, and that distortion transfers on to space adjacent to that, and so on.
The usual visualization for that is a sheet of elastic material. If you depress that material at a single point you get an extended depressed area because every tiny area of the elastic material transfers stress to adjacent areas.
With a fast oscillation you can induce traveling ripples in that elastic sheet. The ripples propagate precisely because the sheet is elastic.
In the particular case of electromagnetism we have that our technology is inducing traveling electromagnetic ripples everywhere. Radio waves are electromagnetic waves.
These traveling ripples keep traveling, independent of whether the original source is still there or not. That is of course very strong corroborating evidence that the electromagnetic field actually exists.
In the case of gravity:
We have the option of thinking of gravity as mediated by a field. This field exist anyway, but in the absence of a source of gravitational interaction this field is thought of as uniform.
A source of gravitational interaction induces a distortion away from that uniform state. This stressed state has an effect on the motion of matter moving through it.
This effect is thought of as a local effect. An object moving through a gravitational field is interacting with the field at the location of that object.
As far as we know, all types of interaction are mutual interactions. When you have two sources of gravity they each contribute to an overall gravitational field, and each object is affected by the gravitational field of the other object.