Yes, we are expected to judge the actions of those inside the church:
1 Corinthians 6:1
If any of you has a dispute with another, dare he take it before the ungodly for judgment instead of before the saints?
But if he will not listen, take one or two others along, so that ‘every matter may be established by the testimony of two or three witnesses.’ If he refuses to listen to them, tell it to the church; and if he refuses to listen even to the church, treat him as you would a pagan or a tax collector.
No, we are not expected to judge the actions of those outside the church:
1 Corinthians 5:12-13a
What business is it of mine to judge those outside the church? Are you not to judge those inside? God will judge those outside.
No, we are not expected to judge the character of those inside or outside the church, only the actions:
Do not judge, or you too will be judged. For in the same way you judge others, you will be judged, and with the measure you use, it will be measured to you.
In other words, we can accuse a Christian of (e.g.) lying if we have evidence, but we can't say, "You lied, therefore you're a bad person". This is what Paul implies when he writes (in Romans 14:4a): "Who are you to judge someone else’s servant?" The context (the weak and strong) clearly implies he is talking about character.
And we should always first look at our own conduct. Perhaps the reason we so easily spot a particular sin in other people is that it also exists in ourselves:
Why do you look at the speck of sawdust in your brother’s eye and pay no attention to the plank in your own eye? How can you say to your brother, ‘Let me take the speck out of your eye,’ when all the time there is a plank in your own eye? You hypocrite, first take the plank out of your own eye, and then you will see clearly to remove the speck from your brother’s eye.