In American Gods, Gaiman appears to have been deliberately vague about how the gods come into corporeal form in their new homeland of America. The only reference I came across is to the dead Mr. Wednesday stating to take shape again in the mists of a cave or something, as a result of Shadow's efforts in passing through the underworld. Then Shadow realizes the truth about his plot with Low-Key and abandons him, and he vanishes back into the mists. I think. I'm probably muddling plot points.
Even in Shadow's discussion with Hinzelmann in the Epilogue, Hinzelmann mentions how he came to America "in the mind of a German immigrant", and a description of how belief in him started east of the Rhine 2,000 years earlier, but I'm still confused as to how he manifested even in his land of origin.
Just to be clear, I'm wondering whether the actual process of manifesting is actually referred to, or was it left up to our imaginations that they sort of 'emerge' into physical bodies, e.g. did Hinzelmann emerge into a solid form as an old man 150 years ago? Or was he born and grew up? Shadow is the son of Wednesday, so HE was born, and appears to develop some mildly godlike powers by the end of the novel (scrubbing people's memories).