Do lured Pokemon appear on sightings?

Solution 1:

No, they do not.

Sometimes this is hard to tell, because there could be another of the same Pokemon nearby, so it shows in your list. However, I have definitely seen Pokemon at lured stops which were not in my sightings list as recently as last weekend.

Here are some other players who have reported the same thing: Reddit Some crazy person that basically wrote a paper on using the sightings list

Solution 2:

No. In addition, Pokémon that are spawning due to incense will no either.

Side Note: Based on screenshots of the beta "Nearby" feature, it looks like Pokémon near Pokéstops will be listed as Nearby in the new system. I assume Lure Module spawned Pokémon will show up in this section, as well. For now, though, lure module Pokémon will not show up in Sightings.