Who is called to judge?

I hear this passage from Matthew,

¹ "Do not judge, so that you may not be judged. ²For with the judgment you make you will be judged, and the measure you give will be the measure you get. ³Why do you see the speck in your neighbor's^a eye, but do not notice the log in your own eye? ⁴Or how can you say to your neighbor,^b 'Let me take the speck out of your eye,' while the log is in your own eye? ⁵You hypocrite, first take the log out of your own eye, and then you will see clearly to take the speck out of your neighbor's^c eye.  -Matthew 7:1-5

quoted often when believers and non-believers complain about Christians judging them. Is this how Jesus meant this parable to be used?

They're right at least that in the context of this passage (and Luke 6:37-42) Jesus is speaking to his disciples so this message was meant for believers and not the Pharisees. I think we can assume that the "log" and the "speck" are meant to represent sin(s); and that trying to remove something from the eye is a type of judging or disciplining. Could the parable mean something else?

Why does the person trying to help his friend have the bigger problem? Why doesn't the one trying to help his friend have the smaller problem? What does this imply?

Does this imply that anyone who tries to judge/discipline someone else will be judged more harshly by God for the sins that they have? This interpretation would be similar to James' command,

¹ Not many of you should become teachers, my brothers and sisters,^a for you know that we who teach will be judged with greater strictness. -James 3:1

Is the "log" that Jesus is talking about meant to represent a specific thing and not be general? Could the "log" be the Jews' failure to recognize the Messiah? I thought this might be the case, but then again the context is Jesus speaking to his disciples.

Paul seems to says something similar in Romans,

¹ Therefore you have no excuse, whoever you are, when you judge others; for in passing judgment on another you condemn yourself, because you, the judge, are doing the very same things. ²You say,^a "We know that God's judgment on those who do such things is in accordance with truth." ³Do you imagine, whoever you are, that when you judge those who do such things and yet do them yourself, you will escape the judgment of God? -Romans 2:1-3

In this context, Paul is addressing church disunity issues and animosity between Jew and Gentile believers.

Let me bring in some more passages. I have been thinking about the question about, "When is the right time to judge?" I think some believers think that the answer is "Never", because we will not be able stop sinning in every area of our life until the resurrection so therefore we will always have a log in our eye. However, Paul calls the church in 1 Corinthians to judge and discipline those in the church who are in sin - specifically a sexually immoral man who was sleeping with his father's wife! It seems that there is a time to judge.

   ⁹ I wrote to you in my letter not to associate with sexually immoral persons-- ¹⁰not at all meaning the immoral of this world, or the greedy and robbers, or idolaters, since you would then need to go out of the world. ¹¹But now I am writing to you not to associate with anyone who bears the name of brother or sister^c who is sexually immoral or greedy, or is an idolater, reviler, drunkard, or robber. Do not even eat with such a one. ¹²For what have I to do with judging those outside? Is it not those who are inside that you are to judge? ¹³God will judge those outside. "Drive out the wicked person from among you." -1 Corinthians 5:9-13

But Paul makes it clear that the church's judgment is only for believers, not unbelievers.

Believers are called to judge other believers, but in righteousness and humility.