Thousands of migrants, mainly from the Rohingya Muslim minority in Myanmar, are thought to be at sea after being abandoned by people smugglers.
About 2,000 of them came ashore in Aceh earlier this week and are being given medical assistance and food.
Since then, Indonesia, Malaysia and Thailand have been turning boats away.
Officials said 210 of those that landed in Aceh on Friday were from Myanmar and 395 from Bangladesh. Medical officials told BBC Indonesian service that eight on board were critically ill.
"According to initial information we got from them, they were pushed away by the Malaysian navy to the border of Indonesian waters," Sunarya, police chief in the city of Langsa where the migrants arrived, told AFP news agency.
He said their boat started sinking after reaching Indonesian territory but local fishermen ferried them to shore, adding they arrived at 5:00 am (22:00 GMT Thursday).
On Friday, a boat carrying about 300 Rohingya Muslims that was stranded off the southern coast of Thailand for a week was sent out of Thai waters.
The BBC's Jonathan Head, who went to the boat near Koh Lipe on Thursday, says those on board had contacted their families to say armed uniformed men had boarded overnight. They repaired the broken engine, gave them food and moved the boat south.
Thai officials said the migrants did not want to go to shore but wanted to continue to Malaysia.
The heavily packed boat was abandoned by people smugglers who also disabled the engine. The migrants, including women and babies, had told our correspondent they had no food or water and that 10 people had died.
Our correspondent said it was a "desperate sight" as people on board cried out for food and water.
'Not a push back'
"We fixed the engine and the boat left last night after 03:00," provincial governor Dejrat Limsiri told AFP news agency.
"We gave them ready-to-eat meals. They are now out of Thailand territory... They will try to go to Indonesia as it seems they cannot get to Malaysia," he said.
Lt Comm Veerapong Nakprasit of the Thai navy said those on board "did not want to come to Thailand so we gave them food, medicine, fuel and water".
"We did our humanitarian duty. They wanted to go to a third country," he told Reuters. "This is not a push back because these people wanted to go."
Our correspondent said the officials taught those on board to operate the vessel before sending them on their way.
'Desperate sight'
Rohingya Muslims have been leaving Myanmar, also known as Burma, because they are not recognised as citizens of the country and face persecution.
Earlier this week thousands of migrants started landing on the shores of Malaysia and Indonesia after being abandoned by people smugglers afraid of a crackdown by authorities in Thailand, the usual route used by the migrants.
It is unclear how many boats full of people are adrift at sea, but rights group say thousands of migrants are likely stranded and are not being allowed to land.
Some of those who are adrift in the Andaman Sea have been provided with food, water and medicine by the Thai navy.
The UN refugee agency has called for countries in the region to co-ordinate a search and rescue operation for some 8,000 migrants it says are trapped on boats in the area.
Some in Washington asked the US government to consider providing humanitarian assistance, and exert pressure on allies in the region to accept more refugees.
"These men, women, children and infants are refugees fleeing well-founded fear of persecution and their deaths may well constitute a mass atrocity in the heart of Asean,'' said Democratic Rep Joe Crowley, referring to the regional block in Southeast Asia that Malaysia, Indonesia, Thailand and Burma are all a part of.
Source: BBC