Most of the world strives for harmony, especially in turbulent war-torn areas. Considering that many of these nations are Muslim nations, it makes sense that leaders of these parts should be the first to take anti-terrorism measures. These nations have the best know-how. It is good news that Saudi Arabia and 34 Muslim-majority nations have formed to fight terrorism with a joint operations base in Riyadh. Syria is agreeing to a ceasefire (even though Syria is not part of the coalition), so there is some progress being made somehow. Other nations will likely take similar action. It seems that nations want the fighting to end. Perhaps the US will also be part of the coalition. The threats will ultimately be lessened, and then nations around the world will be in a position to talk and negotiate for peaceful terms. It'll be harder for ISIS to exert its force when there is a strong opposition in place.
It only makes sense for Muslim nations to lead counterterrorism efforts. They can provide productive examples of how to interact with the world in a nonviolent way. Additionally, terrorist groups are probably more likely to listen to countries that share some values and some religious beliefs, much more so than "secular" nations.
Muslim countries such as Saudi Arabia and Iran which have strong intelligence networks and a well-equipped military are capable of leading the fight against active Islamic fundamentalists. The religious orientation of the governments in these countries also means that they are uniquely placed to influence the beliefs of their inhabitants and Muslims across the world and persuade them away from Islamic fundamentalism.
To say that Muslim nations should lead the counter-terrorism efforts, is to suggest that they are somehow responsible for terrorism in the world. They should support and participate in the fight, but I don't believe that they need to take a leadership role. It is unlikely that western nations would be satisfied with the effort and this may lead to further unrest.
Governments of Arabian nations have been known to support and arm jihad and terrorist groups and they have also been known to keep valuable Intel from the rest of the world. I believe world powers with their very experienced special anti terrorist forces would and should stay in the main fight against terror