How does the whole practice of leaving judging to God work if so many Christians feel they need to judge to show people their errors and that they need Christ?
It depends on what sort of judging we are doing.
When Jesus said "Judge not that you be not judged" (Matt 7:1) a lot of people take that at face value and say we shouldn't judge. But if we read the passage in context it says something very different. Emphasis is mine:
Mat 7:1-5 NKJV "Judge not, that you be not judged. (2) For with what judgment you judge, you will be judged; and with the measure you use, it will be measured back to you. (3) And why do you look at the speck in your brother's eye, but do not consider the plank in your own eye? (4) Or how can you say to your brother, 'Let me remove the speck from your eye'; and look, a plank is in your own eye? (5) Hypocrite! First remove the plank from your own eye, and
then you will see clearly to remove the speck from your brother's eye.
Jesus never told us not to take the speck out of our brother's eye, in this passage often used to teach against judging he clearly says that when we see clearly we can remove the speck from our brother's eye. It's just that the time to do it isn't when we've got a plank in our own eye.
There's also the issue of what Paul wrote about if a brother falls into sin:
Gal 6:1 Brethren, if a man is overtaken in any trespass, you who are spiritual restore such a one in a spirit of gentleness, considering yourself lest you also be tempted.
We can't restore someone who is overtaken in trespass unless we first judge them to be overtaken in trespass. The key here is the spirit of gentleness. It's very easy to look at people and essentially say "I thank you God that I am not like this man here" because of course we would never fall the way they did. But of course there's no way of knowing whether or not we will stumble further down the line.
There are also different forms of judging and testing. Paul told the Thessalonians to "test all things, hold fast what is good" (1Thess 5:21). We can only test something by comparing it against an objective standard and determining whether it measures up. If I offer you a pound of sugar and you're not sure you're actually getting a pound, you put it on a scale to test it. In the same way we can test things against Scripture and determine whether they are sound or not. So it's perfectly appropriate to say "Pastor Joe Blow teaches X is true, Hesitations 4:12-18 teaches that X is false, therefore Pastor Joe Blow is teaching a false message". A counter to that statement could be that Pastor Joe Blow does not, in fact, teach that X is true or that Hesitations 4:12-18 does not conclusively teach that X is false. But to throw around a "don't judge" warning would not be appropriate. If someone were making assumptions about Pastor Blow's motivations in preaching his message they would be on dangerous ground, because we cannot know the heart of another. Pastor Blow may be a wolf deliberately leading people astray, he may be misguided himself, and the chances are we don't know which is the case. To expand "Pastor Blow teaches false theology" with "and I bet he drinks too much and hits his wife" is totally unfounded, and well and truly crosses the line into inappropriate judgment.
As for Christians showing people that they need Christ, Jesus told us to do that. He specifically said to go into the world, preach and make disciples (Matt 28:19, Mark 16:15, Luke 24:46-47). Some Christians aren't very tactful about how they go about that process, but to we can't claim to love someone if we let them rush headlong into a judgment for which they are totally unprepared.
