To debate or not to debate? and When to debate? and How to debate? When we have several people with different understanding of Buddhism, how should we decide who is right? Debate is very tempting in Buddhism, but on the other hand, we must respect other people's views and not create conflicts in the Sangha, so what should we do?

Sometimes Buddhists cannot agree with each other, especially if these Buddhists come from different schools or traditions. An idea that is considered correct in one linage may be considered wrong in another.

Did the Buddha teach how to handle these situations? How were these disagreements handled in the past?

(See also: famous debates in canonical Buddhist texts)

ADDITIONAL QUESTIONS by @Yeshe Tenley:

Under what conditions is it proper? Under what conditions is it improper? Is it a bad thing? Can it be a good thing? What is the proper motivation to do so? What are improper motivations? What is skillful? What is unskillful?

Assuming it is desirable to show respect while debating, how should this respect be manifested? What are examples of disrespectful debate and how can the line be drawn? What's the difference between debate and proselytizing? 

... and references to dharma teachings, historical events, or opinions/literature of noted teachers also welcome and encouraged!
from Kathāvatthu Sutta:


  Monks, it’s through his way of participating in a discussion that a person can be known as fit to talk with or unfit to talk with. If a person, when asked a question, wanders from one thing to another, pulls the discussion off the topic, shows anger &amp; aversion and sulks, then—that being the case—he is a person unfit to talk with. But if a person, when asked a question, doesn’t wander from one thing to another, doesn’t pull the discussion off the topic, doesn’t show anger or aversion or sulk, then—that being the case—he is a person fit to talk with.
  
  Monks, it’s through his way of participating in a discussion that a person can be known as fit to talk with or unfit to talk with. If a person, when asked a question, puts down (the questioner), crushes him, ridicules him, grasps at his little mistakes, then—that being the case—he is a person unfit to talk with. But if a person, when asked a question, doesn’t put down (the questioner), doesn’t crush him, doesn’t ridicule him, doesn’t grasp at his little mistakes, then—that being the case—he is a person fit to talk with.
  
  ...
  
  For that’s the purpose of discussion, that’s the purpose of counsel, that’s the purpose of drawing near, that’s the purpose of lending ear: i.e., the liberation of the mind through lack of clinging/sustenance.
  
  Those who discuss
  when angered, dogmatic, arrogant,
  following what’s not the noble ones’ way,
  seeking to expose each other’s faults,
  delight in each other’s
          misspoken word,
  slip, stumble, defeat.
  Noble ones
  don’t speak in that way.  
  
  If wise people, knowing the right time,
  want to speak,
  then, words connected with justice,
  following the ways of the noble ones:
  That’s what the enlightened ones speak,
  without anger or arrogance,
  with a mind not boiling over,
  without vehemence, without spite.
  Without envy
  they speak from right knowledge.
  They would delight in what’s well-said
  and not disparage what’s not.
  They don’t study to find fault,
  don’t grasp at little mistakes.
  don’t put down, don’t crush,
  don’t speak random words.   


From AN 5.165:


  All those who ask questions of another do so from any one of five motivations.
  Which five?  
  
  One asks a question of another through stupidity &amp; bewilderment. One asks a question of another through evil desires &amp; overwhelmed with greed. One asks a question of another through contempt. One asks a question of another when desiring knowledge. Or one asks a question with this thought,1 'If, when asked, he answers correctly, well &amp; good. If not, then I will answer correctly


Sn 4.8 Pasura Sutta:


  "Only here is there purity"
    — that's what they say —
  "No other doctrines are pure"
    — so they say.
  Insisting that what they depend on is good,
  they are deeply entrenched in their personal truths.  
  
  Seeking controversy, they plunge into an assembly,
  regarding one another as fools.
  Relying on others' authority,
  they speak in debate.
  Desiring praise, they claim to be skilled.  
  
  Engaged in disputes in the midst of the assembly,
    — anxious, desiring praise —
  the one defeated is
    chagrined.
  Shaken with criticism, he seeks for an opening.  
  
  He whose doctrine is [judged as] demolished,
    defeated, by those judging the issue:
  He laments, he grieves — the inferior exponent.
    "He beat me," he mourns.  
  
  These disputes have arisen among contemplatives.
    In them are         elation,
                dejection.
  Seeing this, one should abstain from disputes,
    for they have no other goal
    than the gaining of praise.  
  
  He who is praised there
    for expounding his doctrine
    in the midst of the assembly,
  laughs on that account &amp; grows haughty,
    attaining his heart's desire.  
  
  That haughtiness will be his grounds for vexation,
    for he'll speak in pride &amp; conceit.
  Seeing this, one should abstain from debates.
  No purity is attained by them, say the skilled.  
  
  Like a strong man nourished on royal food,
  you go about, roaring, searching out an opponent.
  Wherever the battle is,
    go there, strong man.
  As before, there's none here.  
  
  Those who dispute, taking hold of a view,
  saying, "This, and this only, is true,"
    those you can talk to.
  Here there is nothing —
    no confrontation
    at the birth of disputes.  
  
  Among those who live above confrontation
    not pitting view against view,
    whom would you gain as opponent, Pasura,
  among those here
  who are grasping no more?  
  
  So here you come,
    conjecturing,
  your mind conjuring
    viewpoints.
  You're paired off with a pure one
    and so cannot proceed.  


Most people approach debate as if it was a sword fight. They debate to win, as if losing meant death. They do not make any effort to understand the other person's perspective!

They keep on fighting until they see an opening in the opponent's defense. As soon as they see a little inaccuracy or a logical error, they happily grasp on that -- even if it does not change the main idea the other person is trying to convey.

When the other person gets confused for a moment and says something obviously wrong, they happily make fun of him and keep pushing until the opponent is completely lost.

This kind of people firmly hold their ground, their confidence coming from their limited perspective and narrow view. Having won another debate with aggression and cunning, they are proud of themselves.

But if they see they cannot win, they get very envious of the winner, they get nervous, start changing topics, lie, resort to personal attacks, get angry, and finally go down to rudeness and yelling.

Instead, the noble students do not debate, they have conversations. They discuss things. When they thus sincerely talk they try to see each other's perspective. They understand that each (truthful) perspective is valid from its own side. So they try to see the value of the other perspective, and appreciate the factors and conclusions of the other side. Then, by combining the two perspectives, they come out with a more complete, more thorough understanding of the original problem. Each side lets go of its attachments and delusions as it accepts the other side's point of view, and integrates them together.

Also, see MN18 for an in-depth analysis of the psychology of reification underlying debates and arguments.