There is probably no other bird on this planet that comes more closely to what could be called a cryptid than the Manumea, the Tooth-billed Pigeon (Didunculus strigirostris (Jardine)) of Samoa.
The species is known to inhabit, or at least to have inhabited, the rainforests of the islands of Nu’ulua, Savai’i and ‘Upolu, Western Samoa; in prehistorical times it was even more widespread. Nearly nothing is known about this species: the breeding behavior is still unknown, the same more or less applies to basically all of the bird’s habits.
As far as I know there are only about five or so photos of living individuals of the species, most of them, if not all, show the same bird that was kept in captivity for some time.
The latest sightings were of a juvenile bird in 2013, which also was photographed; than a bird was seen and heard calling in 2020, however, no photo had been taken this time. 
The Manumea is currently not kept in captivity and the wild population is estimated to be less than 100 – to about 300 birds, that’s not much and the species is in immediate danger of extinction.
Wouldn’t it be phantastic if even only a fraction of the amounts of money that are spend to prove the existence of such phantasy creatures like Bigfoot, Chupacabra or Mokele Mbembe would be used for something useful, for the search for the Manumea, for the rescue of this enigmatic yet indeed existing bird?!
The genus Pareudiastes consists of two species that are known from historical times, meaning ‘having been seen’ by western scientists, we can probably add at least two undescribed extinct forms that are known exclusively from scanty subfossil remains, one from Fiji and one from the Solomon Islands.
The two species of which at least skins remain are very little known, the Puna’e (Pareudiastes pacificus Hartlaub & Finsch) from Samoa in fact is the best known of them.:
The Puna’e, whose name roughly means ‘springs up’, is known to have inhabited the rainforests of Savai’i, Samoa, it is said by the natives to have lived in burrows which were longer than a man’s arm and which ended in a sort of chamber in which the bird slept during the day.
The large eyes of the species indeed point to somewhat nocturnal habits.
When the bird was disturbed it jumped up from its burrow with fluttering wings, but being flightless it landed shortly after and run away quickly.
It is furthermore known that it was not a vegetarian species, since it died when it was fed with plant material but was ‘happy’ when fed with insects.
There is at least one reliable account that indicates that this species also inhabited the neighboring island of ‘Upolu.:
“The Samoans always speak of the Pareudiastes as the ‘bird which burrows like a rat.’ Again and again when I have put the question to a native, ‘Do you know the Puna’ e?’ the reply has been, ‘No, I have never seen it; but that is the bird of which the old people speak that it used to be very plentiful long ago, and that it burrows like a rat and lives underground.’ It is very rarely that I have met with any one who has seen the bird; but I have met with two persons who have actually taken it in its burrow. The first is a man well known to me, and in whose veracity I have faith. He says that about four years ago [ca. 1870] he was one of a large party hunting feral pigs in the mountains of Upolu, when they came upon a burrow which one of the party pronounced to be the hole of a Puna’e. My informant says that he put his arm into the hole, and at its extremity (which he could barely reach) he found the bird. He drew it out, and, taking it home, tried to tame and feed it; but it would not eat, and soon died.” 
Yet, how is this possible? The islands of Savai’i and ‘Upolu are separated by the 13 km wide Apolima strait.:
During the Pleistocene the sea level was lower and Manono very likely was connected with ‘Upolu, but Apolima was not, and Savai’i and ‘Upolu also weren’t connected.
So, how did a flightless bird manage to get from one island to the other?
Well, the answer might be that the Puna’e wasn’t flightless at the time when the sea level was lower, or that the birds from ‘Upolu represented a distinct (sub)species.
What do we know about the second historical known species, the Makira Woodhen (Pareudiastes silvestris (Mayr))?
Well, not that much.:
This species is known from a single specimen that was taken in 1929 on the island of Makira, Solomon Islands in montane forest at an elevation of about 600 m, only its skin was preserved, the bones not, and it apparently was flightless or at least nearly so.
The natives called it Kia and hunted it with dogs.
The species was apparently still ‘well-known’ by the natives in 1953, and they also said that it was not rare, nevertheless not a single one was ever seen since.
The Makira Woodhen, or Kia, however, is the sole member of this genus that may in fact still survive, and I personally hope that it might be rediscovered some day.
The other two additional forms, as i said before, are known only from some scanty subfossil remains found on the island of Buka in the northernmost part of the Solomon Islands as well as on Viti Levu, the largest of the Fijian Islands respectively.
There’s not much that can be said about these two, so let’s hope that some day more remains are found to shed some light on that matter.
 Letter from Rev. S. J. Whitmee. In: Proceedings of the Zoological Society of London 1874: 183-186