It is only briefly touched on in "The Search" but part of shapeshifting as it relates to the Founders is not simply assuming the shape of a certain thing, but actually "becoming" that thing.
So Odo sees just as anyone else would, the only difference is he has to make the eyes and ears, and presumably some kind of rudimentary central nervous system to process it all.
Besides that, it's said the Founders were once solids, so it's likely that somewhere in their collective consciousness is the "blueprint" for humanoid form and probably other forms of life. Remember Odo was drawn to "becoming" life forms, during The Search he says he had never contemplated just hanging out as a rock. Considering his mission it was likely encoded into his DNA to make shapes that would be able to interface with the other life he encountered.