This comes from Greek en, in, plus gaster, belly, plus muthos. speech, so it is the exact equivalent of Latin ventriloquist, which comes from venter, belly, plus loqui, speak.
Engastrimyth is now hardly common (indeed it’s not in most current dictionaries) and on the rare occasions that it appears is never applied to the much debased “gottle of gear” entertainer’s chat with a dummy. Instead, it refers to the classical soothsaying phenomenon of speaking without appearing to speak, associated especially with prophetesses such as the famous Delphic Oracle, or with seers who acted as conduits for the voice of someone beyond the grave, such as the Biblical story of the Witch of Endor.
Belly seems to have been something of a euphemism, in fact, since some writers thought that the voice came from the genitals. From ancient times up to the eighteenth century, the phenomenon was usually linked to religious frenzy or demonic possession, though it was frequently denounced as fakery. It was only at the end of the eighteenth century that ventriloquism became a form of entertainment and it seems to have been rare that engastrimyth was applied to it — then usually only in rather formal literary contexts.