SNP leader Swinney told Radio Wales Breakfast it was a “significant and pivotal moment for the United Kingdom”, and that things had “changed forever”.
“The people of Scotland, Wales and Ireland have a right to self-determination,” he said. “They are entitled to decide on their own future.”
“Fundamentally what our discussions today are about are about recognising the significance of the moment we are in.”
Sinn Fein vice president O’Neill said the best interests of people across the island of Ireland “have never been served by those in Westminster”.
“However, change is happening right across these islands and we must be ready for constitutional change when it comes,” she said.
“The election of three nationalist first ministers for the first time on these islands is a clear indication that people see that Westminster is not working for the people of the North, Scotland or Wales.”
Mary Lou McDonald, president of Sinn Féin, told Wales the trajectory for Wales, Scotland and Ireland to self determination was different, and that we were living through “history-making times”.
“Obviously we have different national stories, but we have one very critical thing in common and that is our absolute insistence that it is for the people of Ireland to decide the future of Ireland, for the people of Scotland to decide the future of Scotland and for the people if Wales to decide the future of Wales,” she said.
The meetings come days after the matter of Scottish independence resurfaced in the Commons.
During Prime Minister’s Questions on Wednesday, Andy Burnham appeared to indicate a second referendum could be held in Scotland if there was clear public support for a vote.
He compared the situation to the conditions for a border poll in Northern Ireland, explaining the Good Friday Agreement made clear a poll would be “considered by the Northern Ireland secretary when there is a clear consensus in the country, when public opinion has changed to the point where the call has to be considered”.
“I would say that it is exactly the same situation in Scotland,” he added.
The apparent softening in tone was welcomed by Swinney before Burnham clarified that another Scottish referendum remained “off limits”.
A referendum on Scottish independence in September 2014 saw the country vote to remain part of the UK by a margin of 55% to 45%.