Can a "static" magnetic field carry Information?
A static field (indeed, any static symbol) conveys no information, by
Shannon's theorem.  This is because the probability of the state is
1, there is only one 'symbol', and if it is unchanging, the frequency
bandwidth is zero.  

Once one talks of amplitude modulation, however, the field is NOT static,
there is a bandwidth, and a signal that can carry information.
Every loop-shaped antenna picks up modulated magnetic
field, so we know that kind of receiver is workable.
In a sense, a permanent magnet with a shutter (driven vanes of soft
steel, perhaps?) makes such an amplitude modulated signal.   Such a
modulator, in my auto, senses crank position from the passage of
a notched rotor next to a magnet and sense coil.

The crank sensor might be less sensitive if electrically shielded, but
would still work.   Magnetic fields are hard to shield.