Disagreements are necessary for growth of an individual or any organisation. However, growth will only result if disagreement leads to healthy discussion to reach a common ground.

Unfortunately, individuals get so emotionally attached with their ideas that not many take kindly to any dissension. There are also individuals who express dissension just score to brownie points or let down an ‘idea originator’. It is for these reasons that most disagreements lead to violent debates with no fruitful results.
Disagreements, if not handled maturely, will spoil professional/ personal relations. I have seen friends / young couples breaking up as they are not able to handle disagreements.
How to make disagreement more meaningful….
Debates can only be meaningful if the participants have in-depth knowledge of the subject they are discussing and are open to accepting new ideas.
The tone of conveying a disagreement should not be humiliating. How one speaks matters as much as what is being spoken. The aim should be to oppose the idea and not the originator. Sarcasm and humiliation will only lead to violence or one party just withdrawing. In both cases the organisation is bound to suffer.
I have seen many a seniors out rightly humiliating a junior who happened to voice dissension. Thus, forcing the junior to just step back. It may be better at times to avoid expressing the dissent publicly. Sometimes what is not acceptable in a conference room may be acceptable to the boss in the confines of his office.
People should rely on the depth of their knowledge instead of the weight of their appointments/ connections to push through their idea.
Do not disagree to gain attention/ importance or let down an individual. This usually happens when two individuals are competing for the same space.
Do not be in a hurry to oppose. Allow the speaker to say whatever he must. There may not be a need to oppose at the end of the presentation.
Disagreement should rest on facts and not based on emotions/ feelings.
The opponents should understand that discussion cannot be unending. There will be a point when it will start losing steam/ meaning. That is the time they should gracefully withdraw. Discussions are generated to find better solutions and not as a means for personal wins.
Conclusion….
The organisation should provide an environment which supports constructive disagreements. This will encourage employees to take risks and improve the system.
The employees should also understand that disagreements are a tool for improvement and not a means to score points.