In a statement ahead of the Moscow meet, the network said they would not accept peace at the cost of their freedoms and urged delegates to defend the rights of half of Afghanistan's 35 million people.
The two-day meeting in the Russian capital kicked off Tuesday. "It's very positive, good news".
On Tuesday, senior Afghan officials, including former President Hamid Karzai, met with Taliban representatives in Moscow weeks after the insurgents met with USA representatives to discuss an end to the war.
He said his administration was holding constructive talks with a number of groups, including Taliban militants.
"We do not know whether we will achieve an agreement - but we do know that after two decades of war, the hour has come to at least try for peace", Trump said in his State of the Union address.
"Today a lot of Afghans sit together, we believe InshaAllah (God willing) after the withdrawal of foreign forces, after the withdrawal of occupation forces, we can sit together and we can resolve all issues together".
Lead Taliban negotiator Sher Mohammad Abbas Stanikzai said on Tuesday the movement did not want to rule alone but as part of "an intra Afghan Islamic system of governance.in consultation with all Afghans".
A 10-man Taliban delegation said it hoped to use the talks for "opening channels to reaching an understanding with non-government Afghan political groups".
The Taliban, who brutally ruled Afghanistan from 1996 to 2001, has always refused to break bread with Ghani and Kabul, who they view as U.S. stooges.
"We understand that the government needs to be part of these negotiations". So the constitution would be Islamic, there would be no doubt - it would be for a long time and it would be after this also. THey were armed with modern weapons to launch quick attacks, he added.
"We are exchanging our views". The conference is convened by the Afghan diaspora and is not a follow-up to the Moscow format on settling the crisis in Afghanistan, the latest meeting of which was held in November 2018.
The departure of USA forces - if achieved under a deal with the Taliban - could draw the region into a "new war" for dominance in Afghanistan.
Stanekzai said the Taliban also has a position on women's rights.
"They said women can work, women can be educated, women cannot be presidents though - they can be prime ministers".
"Let us not speak about those things".
"I think all sides are ready for a compromise".
Ghani's allies in Washington insist Afghans should lead the peace process, and ostensibly the months-long push by the United States to engage the Taliban has been aimed at convincing them to negotiate with Kabul.
"There's something surreal about Afghanistan's most powerful opposition politicians meeting with the Taliban in order to spark an intra-Afghan dialogue that doesn't include the country's own government", analyst Michael Kugelman of the Wilson Center in Washington told AFP. Where is their executive power? Anyone without formal registration to enter the hotel was not allowed on the premises, and black sedans with Russian license plates and single roof-mounted flashing blue lights - the sign of Kremlin or State Duma officialdom on Russia's roadways - were seen coming and going from the hotel entrance.
Ghani's allies in Washington insist Afghans should lead the peace process, and ostensibly the months-long push by the USA to engage the Taliban has been aimed at convincing them to negotiate with the government in Kabul.
Ghani has repeatedly said that all Afghans should agree on the need to end hostilities and an eventual withdrawal of foreign forces, but that he would not "surrender to a temporary peace deal".