INSIDE THE FAMILY OF GOD - HOLDING EACH OTHER ACCOUNTABLE
It is quite clear the answer to the question is "yes." We are called to humbly and yet decisively hold one another accountable within the family of God, within the company of the redeemed.
First, Jesus gave his followers a process for dealing with sin which occurs between them. In Matthew 18 the Lord instructed,
"If your brother sins against you, go and tell him his fault--between him and you alone. If he listens to you, you have gained your brother. But if he does not listen, take one or two others along with you, that every charge may be established by the evidence of two or three witnesses. If he refuses to listen to the witnesses, tell it to the church. And if he refuses to listen even to the church, let him be to you as a Gentile and a tax-collector."
This process could hardly go on without some evaluation or discerning, some "judging" (if you will). Of course, this must be more than a "he said" vs. "she said." Witnesses of the wrong should be brought in, when necessary, to make verify objectively that sin has been committed and needs to be confessed and left behind.
Realizing personal interests can become selfish and harmful to others, Jesus outlined a simple way of keeping our accounts clean and clear with one another. His method implies that people will trust each other's appraisals and respond humbly to one another. Defensiveness will only aggravate and create distance.
Second, Paul asks the Corinthians to evaluate (or judge) a man in their congregation who was openly living in sexual sin, "having his father's wife" (cf. 1 Corinthians 5). He was someone who claimed to trust in Christ and follow Christ, but saw no need to step out of a relationship that was even shameful among unbelievers (i.e., incestuous). Within this family of God, such open, sinful, and unholy behavior was not to be tolerated. It shamed the reputation of God's holiness among His people. Within this family of God, there was to be a call is man's life to step away from this relationship, repent, and honor the Lord and His Word. Should he refuse to do so, he was to be set outside the fellowship of believers.
Paul was embarrassed that the believers had tolerated such open sexual sin for so long. Some were even arrogant about it, rather than mourning over it. "I wrote to you in my letter not to associate with sexually immoral people--not at all meaning the sexually immoral of this world, or the greed or swindlers, or idolaters, since then you would need to go out of the world. But now I am writing to you not to associate with anyone who bears the name of 'brother' if he is guilty of sexual immorality or greed, or is an idolater...Is it not those inside the church whom you are to judge?" (1 Cor 5:9-12).
So the family of God should humbly yet firmly deal with those who claim the name of Christian but want to live in open, unchecked sin. Evaluation and clear consequences must follow for anyone who wants to claim Christ but live by Satan's license.
Third, there are a number of examples where fellow disciples needed to rebuke other disciples in order to urge obedience to the Lord and His Word. Paul called out Peter for being a hypocrite with the Gentile believers in Antioch (Galatians 1). Paul urged the believers in Philippi to help two women sort out their disagreements and disputings (Philippians 4). James had some straight to those who were not demonstrating true faith and religion by refusing to help those in need (James 1). He urged that Christians "confess sins to one another and be healed" of spiritual discouragement (James 5).
CONCLUSION - the Family of God humbly judging one another
All of these things call for a "judging," that is, a discerning of what behavior dishonors the Lord and must be corrected. Christians are to humbly look to themselves first, but also be willing to hold one another accountable for a living that pleases God. Again, Paul urged this mutual sharpening when he wrote, "Admonish the idle, encourage the fainthearted, help the week, be patient with all men" (1 Thessalonians 5:14).
In these ways -- when we are humble with and responsive to one another -- do we become more like our Savior.