Why is the matter to be first kept private between you two? Because our love for our brother requires it. If we are sincerely committed to love for our brethren as Jesus commands (John 13:34), then even if a brother has sinned against us or has done us wrong, we will not want to maliciously injure that brother's credibility within the body — that might hinder his restoration. If this person's transgression against you is circulated within the body, but later he repents, many in the body will have already judged this brother and the accusations will have damaged their opinion of him. By spreading our accusation against an offending party, it builds a consensus against them and makes it difficult, if not nearly impossible, to restore that person to the body should they repent and desire to make things right.
Also, many alleged trespasses between brethren are a result of "misunderstandings." We need to first investigate the facts and find out for sure whether a trespass has really occurred. This is another reason why you are to first go privately to the brother or sister in question — to confront them with the alleged offense and hear their side of the story. If we disclose the offense to friends in the body, we may later discover that the matter was only a misunderstanding. But by then, their reputation will have been damaged by our allegations.