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So I poked around the literature and it says (this is from 2013):

    Behavioral evidence indicates that there are magnetoreceptors in the beak of birds. These receptors include magnetite, as indicated by the pulse experiments, and they mediate their input to the brain by the ophthalmic nerve and the trigeminal system. They are not involved in the avian magnetic compass; instead, they seem to normally convey information on magnetic intensity. Their natural function appears to be to provide birds with magnetic information as one factor in the multi-factorial navigational ‘map’—not only homing pigeons within their home region, but also migrants when they return to their familiar breeding site or wintering area. The exact position of these magnetite-based receptors is unclear. The effect of the local anesthetic seemed to speak in favor of the receptors described in the skin of the upper beak (e.g., Hanzlik et al. 2000; Fleissner et al. 2003; Falkenberg et al. 2010), yet the histological study by Treiber et al. (2012) calls the existence of these receptors in question, a finding that received considerable public attention. This may point to the single-domain receptors described in the nasal region (e.g., Beason and Nichols 1984, Beason and Brennan 1986; Williams and Wild 2001), but it appears highly unlikely that the externally applied anesthetic could have reached them. The observation that young chickens with the tip of their beak removed, as routinely done in the poultry industry, were impaired in locating a magnetic anomaly (Freire et al. 2011) also suggests a position of the receptors further in front of the beak.

The first thing that jumps out at me is that the trigeminal nerve (specifically the ophthalmic branch) and the trigeminal system in general mediate this. The trigeminal nerve handles both sensory and motor signals, and the histology doesn't seem to line up with touch receptors observed elsewhere in bird bodies.

Wikipedia sez:

    For animals the mechanism for magnetoception is unknown, but there exist two main hypotheses to explain the phenomenon.[4] According to one model, cryptochrome, when exposed to blue light, becomes activated to form a pair of two radicals (molecules with a single unpaired electron) where the spins of the two unpaired electrons are correlated. The surrounding magnetic field affects the kind of this correlation (parallel or anti-parallel), and this in turn affects the length of time cryptochrome stays in its activated state. Activation of cryptochrome may affect the light-sensitivity of retinal neurons, with the overall result that the bird can "see" the magnetic field.[5] The Earth's magnetic field is only 0.5 Gauss and so it is difficult to conceive of a mechanism by which such a field could lead to any chemical changes other than those affecting the weak magnetic fields between radical pairs.[6] Cryptochromes are therefore thought to be essential for the light-dependent ability of the fruit fly Drosophila melanogaster to sense magnetic fields.[7] The second proposed model for magnetoreception relies on Fe3O4, also referred to as iron (II, III) oxide or magnetite, a natural oxide with strong magnetism. Iron (II, III) oxide remains permanently magnetized when its length is larger than 50 nm and becomes magnetized when exposed to a magnetic field if its length is less than 50 nm.[8] In both of these situations the Earth's magnetic field leads to a transducible signal via a physical effect on this magnetically sensitive oxide.

The answer appears to be 'we have no fucking clue'.

In cartilaginous fish, there are the ampullae of Lorenzini:

    These organs are made up of mucus-filled canals that connect from the skin's pores to small sacs within the animal's flesh that are also filled with mucus. The ampullae of Lorenzini are capable of detecting DC currents and have been proposed to be used in the sensing of the weak electric fields of prey and predators. These organs could also possibly sense magnetic fields, by means of Faraday's law.

Hens have mineral deposits in the dendrites in their beak, which is radically different from the usual touch receptor.