I just read through Jude and the section in 2 Timothy you brought up. I wondered how it is that we "contend for the faith", "correct", "rebuke", "reprove error in person's teachings and lives".
After pointing out that false teachers turn the grace of our God into lasciviousness, and denying the only Lord God, and our Lord Jesus Christ (Jude 1:4), and describing the activities of these false teachers, Jude tells us:
Jude 1:
20 But ye, beloved, building up yourselves on your most holy faith, praying in the Holy Ghost,
21 Keep yourselves in the love of God, looking for the mercy of our Lord Jesus Christ unto eternal life.
22 And of some have compassion, making a difference:
23 And others save with fear, pulling them out of the fire; hating even the garment spotted by the flesh.
It appears that we are to:
1) build ourselves up in faith, pray, keep ourselves in the love of God, look for mercy
2) have compassion on those who fall into the hands of false teachers
3) some we may have to pull from the fire
As far as the section in 2 Timothy 4, we really need to go back to 2 Tim 3 where we are told that all Scripture God breathed and is profitable for doctrine, for reproof, for correction ... that the man of God may be fully equipped unto all good works (2 Tim 3:16-17). Then ...
2 Timothy 4:
2 Preach the word; be instant in season, out of season; reprove, rebuke, exhort with all longsuffering and doctrine.
3 For the time will come when they will not endure sound doctrine; but after their own lusts shall they heap to themselves teachers, having itching ears;
4 And they shall turn away their ears from the truth, and shall be turned unto fables.
5 But watch thou in all things, endure afflictions, do the work of an evangelist, make full proof of thy ministry.
It appears to me that in both Jude and Timothy, we are to help those who have fallen into the hands of the false teachers ... reach out to them and bring them back to the truth of God's Word.
And 2 Tim 2:4 tells us to be instant in season, out of season ... which is pretty much all the time. Reminds me of that verse in Peter where we are told to be ready always to give an answer to every man that asketh you a reason of the hope that is in you with meekness and fear (1 Peter 3:15).