Debate plays an important part of Buddhism. What are famous debate encounters by the Buddha and also by other Buddhist masters like Dharmakīrti, especially with Hindus(exponents of Brahmavāda-Ātmavāda) and Jains?
A really good look at the sort of debates Buddhists got into is in the Kathaavatthu - translated into English as "Points of Controversy". This is a later text, but outlines the many controversies and associated arguments. Examples of good debates in the suttas are MN 56 and DN 23.
As you say, debate plays an important part in the teaching, so the debates are really too numerous to list. I don't have any information on Dharmakirti though.