The Polynesian Ground-Dove (Pampusana erythroptera (Gmelin)) and the Friendly Ground-Dove (Pampusana stairi (Gray)) are mentioned in several enumerations of birds that are thought to inhabit the Gilbert Islands in the Micronesian part of Kiribati.
So, I did a little search to find out something more ….:
The first thing I found was that D. H. R. Spennemann and H. Benjamin name these two species as inhabitants of the Marshall Islands, which apparently is an error since they use A. B. Amerson’s account as their source, in which they are clearly named as inhabiting the Gilbert – but not the Marshall Islands. .
This is the abovementioned account by A. B. Amerson Jr..:
“Gallicolumba erythroptera* Ground Dove
Status — Introduced breeder in the Gilbert Islands.
Marshall-Gilbert Distribution Records — Breeding: Gilberts – Abemama, Nonouti.
Pacific Distribution — Native to the Society and Tuamotu Islands (Baker, 1951).
*Remarks — I am using the species designation given by Child (1960), who says this species was reported to have been introduced into the Gilberts from Nauru about 1940. Pearson (1962) did not find any doves at Nauru in 1961.” 
The Gilbertese name of this bird is given here as bitin, more about that later.
“Gallicolumba stairi* Friendly Ground Dove
Status — Introduced, possible breeder in the Gilbert Islands.
Marshall-Gilbert Distribution Records — Gilberts – Abemama.
Pacific Distribution — Present only at Fiji, Tonga, and Samoa Islands (Peters, 1934)
*Remarks — I am using the species designation given by Child (1960) who says this species was probably introduced from Fiji.” 
Here the author states “Child (1960)” as his source, so let’s see where that leads me to.
I could trace all accounts back to the following one, which appears to be the first that mentiones these two dove species, it is from a Peter Child who worked for about one year as an radio operator in the Coastwatching Organisation during WWII and who visited nearly all the islands in the ‘colony’ within a time of three years from 1953 to 1956.
He was an amateur bird-watcher, as he called himself, but his accounts are in fact reliable, in my opinion.
Here they are.:
“24. Gallicolumba erythroptera: Ground Dove.
Lupe palangi. Bitin (Taobe)
While Ducula is a native in the Ellice, being referred to in many old songs and legends, the ground doves are undoubtedly recent introductions of the present century, probably mainly from Fiji. At Abemama they are reported to have been introduced from Nauru about 20 years ago [ca. 1936 because this account was originally puplished in 1956], and have multiplied considerably so that there is now a fair number in a wild state. A few pairs have also been taken from Abemama to some other Gilbert Islands as pets, but in most of the Colony they are unknown. A pair taken to Nanouti had four offspring, and in June two females had nests about ten feet of the ground in an old deserted house; the nests were of grass and stra and built inside old boxes; each contained two eggs, oval in shape and creamy-white in colour. At Abemama they are said to nest in coconut crowns, often high above the ground. They feed mainly on the ground; when disturbed they fly up into the palms and their call, a soft “coo”, may then be heard.
The colouring is typically darkish greys and white; the head, neck, back and upper breats are grey with a purplish and greenish sheen or irridescence; the secondary wing feathers are mainly dark grey and the primaries and tail feathers mainly white; the abdomen is white, often speckled with grey, and he under tail-coverts white. The short bill is dark grey with a small whitish operculum at the base; the legs and feet are coral red, or purplish red. Some birds have less white than others, and hardly any two are exactly alike.” 
The name Bitin apparently is the local variation of the English word pigeon, while the name Taobeundoubtedly is the local variation of the German word Taube which means nothing but dove/pigeon; both these names suggest that the birds were imported from somewhere else.
The name Lupe palangi is the Tuvaluan name given to this bird, it can be translated as foreign pigeon (!).
It also appears to me that he did see this species with his own eyes.
He goes on with the second dove species.:
“25. Gallicolumba stairii. Friendly Ground Dove.
There are few individuals in a semi-wild state at Abemama, probably of this species, and probably introduced from Fiji where it is common. The habits are similar to those of G. erythroptera, and there is no distinction in vernacular names of the two species.
The colouring is mainly brown with a little white on the wings and lower breast; the upperparts have a greenish sheen in some lights. The bill is dark and the feet deep red or purplish red.“
All in all it appears to me that both, the Polynesian- as well as the Friendly Ground Doves, indeed had been introduced sometimes during the middle 1930s, during WWII to at least the Abemama atoll, it appears not to be known by whom, and it also appears not to be known if they still exist there.
Interesting is the statement that at least one of them had been brought from Nauru, where no groud dove species is known to have existed – and if one has existed it might rather have been a distinct, endemic one, but not the Polynesian Ground Dove.
Maybe someone with an interest in birds will be able to make a visit to the Abemama atoll some day and clarify these questions.
 P. Child: Birds of the Gilbert and Ellice Islands colony. Atoll Research Bulletin 74:1-38. 1960
 A. B. Amerson Jr.: Ornithology of the Marshall and Gilbert Islands. Atoll Research Bulletin 127: 1-348. 1969
 Robert P. Owen: A checklist of the birds of Micronesia. Micronesica 13(1): 65-81. 1977
 Dirk H. R. Spennemann; Hemley Benjamin: Notes on the Avifauna of Ebon Atoll, Republic of the Marshall Islands. Historic Preservation Office 1992