What can be measured or derived about a remote magnetic field? This question related to Why are magnetic lines of force invisible? and is motivated by a comment of @BlackbodyBlacklight, based on that, the illustrating example may depend on that linked question as context to be clearly understandable.
A remote magnetic field, in the sense that it is not at the location of measurement, could influence the location of measurement in some (possibly indirect) way that allows to derive information about it's structure.
This is comparable to deriving information about a remote temperature profile based on properties of the local electromagnetic field, like when using a camera, or just seeing something glow.
It might well turn out that it is fundamentally impossible to derive information about a remote magnetic field, (given some sensible constraints).
In this case, an Answer should ideally explain why that is the case.
What is described above is roughly comparable to human perception, which was the context where the question came up originally. Therefore, I will illustrate my initial ideas in that context in the section below:

Establishing the context for the question (The biological aspects referred to are part of the illustration, not directly related to the question):
The motivating idea was: "We can not see magnetic fields, but that may be because it was not important during evolution to acquire this capability."
Could it be possible, in principle, to "see" magnetic fields?
Now, if it would have been helpful during evolution - what kind of 
perception is possible purely from the physical side of the question - assuming "perfect evolution".
The linked question asks about seeing magnetic field lines - so could something like eyes for seeing field lines have evolved?
I assume not, so we do not need to go into details whether to see them on surfaces, as lines at a fixed distance, etc. (Feel free to make creative assumptions as needed regarding how to "see")
What did evolve, in some birds and bacteria, is perception of the field of Earth in terms of direction of the local(!) field lines - something like "feeling north and south".

The actual question, related to physics of magnetic fields, in comparison to phenomena for which human perception exists:
What are the physical constraints?  
Seeing a magnetic field like a fourth base color would not work - there is no radiation.
Something similar to spacial sound perception? Which would mean to measure from a finite set of "local" reference points to collect measurements on a given remote locatioin.
Anything better than measuring a local field vector is certainly interesting.  
 A: Sensing any kind of wave requires that it either impinge on the observer, or produce a local, temporary side-effect such as a phase change, which is then visible by affecting the propagation of other waves which impinge on the observer.

A wave generates pressure, causing temporary condensation, wherein water droplets reflect visible light.

A wave affects its medium's density, altering the index of refraction and generating caustics as directional light passes through.
A sufficiently powerful magnetic field will ionize nearby molecules, due to local magnetic moments and those of elementary particles. That should produce a visible effect, although being close enough to see it would probably be lethal.
On a friendlier scale, iron filings trace magnetic field lines to make them visible, but this doesn't reveal the field strength very well. Some filings may stand on end where the field isn't perpendicular to the surface containing them, and it has sufficient strength.
Being able to sense the value of a field at a remote point without any intermediary would be "spooky action at a distance."
A: Good question and interesting discussion. But i am afraid it is still possible :) 
Theoretically you can can have an alien bionic/machine creatures who have  particle  accelerators (mini/nano etc.) inside of them. This accelerators can be suited to see magnetic field almost similar way as with vision. BUT. Funny enough in this case it seems not possible to LOOK outside but only inside of the creature. Some kind of inner vision. Or. If the creature is allowed to constantly emit particles outside of it then also outside vision can be possible.
Off course this is just a sketchy guess. Details need to be elaborated.
