The authoritarian leader took power in a 1994 coup and stepped down overnight in the face of pressure from West African armies that entered Gambia to force him to recognise that he lost an election in December to President Adama Barrow.
But a top Barrow adviser, Mai Ahmad Fatty, said the declaration did not have the new leader's approval.
On behalf of the Coalition for Reforms and Democracy and Opposition fraternity, I extend congratulations to President Adama Barrow as he formally takes over the management of his country after the crisis following presidential elections last month.
The declaration stated that Jammeh's departure from The Gambia on Saturday was temporary adding that it was in order to assist a peaceful and orderly transition and transfer of power and the establishment of a new government.
Jammeh lost the presidential election in December a year ago to Adama Barrow but said he would not relinquish power, claiming that there had been irregularities in the vote and pushing the country to the edge of war.
"President Jammeh and his team concocted a declaration to be endorsed by (regional bloc) ECOWAS, the United Nations and the African Union that gave him every guarantee, essentially impunity", Mankeur Ndiaye said.
The choice of Equatorial Guinea for his exile helped ease concerns that Jammeh might interfere in Gambia's politics if he stayed in Guinea, whose border is not far from The Gambia's eastern region. "We are talking about getting a truth and reconciliation commission", he said.
The Gambian political crisis
Worldwide rights lawyer Reed Brody said the declaration "doesn't give him an amnesty, and under global law in fact you can't amnesty certain crimes like torture and massive or systematic political killings".
Amnesty International issued a statement that said Gambia would become a model of democracy and that its citizens can now stop living in fear. Shops and restaurants opened, music played and people danced in the streets.
The official also said Barrow must be in place "as soon as possible".
As he waited for the arrival of the regional force in Banjul, Gambia defense chief Ousmane Badjie said the military welcomed it "wholeheartedly".
Some of the 45,000 people who had fled the tiny country during the crisis began to return.
The focus for Gambians has now turned to calling Jammeh to account for some of his alleged human rights abuses.
Rights groups say those include arbitrary detentions, torture and even killing of opponents.