The situation had put further strain on ties between the two Horn of Africa nations, who fought a 1998-2000 border war and are blaming each other for stirring up the Somalia conflict.
Eritrea denies involvement
Ethiopia said the government in Asmara was behind the abduction.
"There is no doubt the abduction was masterminded and executed by the Eritrean government," Ethiopia's ministry of foreign affairs said in a statement last month.
"It is evident the Eritrean regime not only supports and propounds terrorism, but it is also directly involved in active terrorism."
A regional police chief claimed that the abduction had been carried out by men in Eritrean military uniforms.
Eritrea has repeatedly denied any involvement in the incident.
Analysts believe a separatist group from the region, which straddles both countries, was behind the kidnap.
Afar separatists began a low-level rebellion against the Addis Ababa government in the 1990s, calling for the creation of a separate Afar state on territory which straddles Ethiopia, Eritrea and Djibouti.