How much realization of one's own sinfulness is needed for a person to be saved? I have met some Christians who seemed to put a lot of emphasis on their realization of how sinful they were. Others, however, didn't seem to be stressing this point that much.

Honestly, in my personal experience, the joy of knowing the Lord, the joy of the very first prayer to Christ, and the joy of having fellowship with Christians came first. The first true realization of how sinful I am came much later. In fact, this realization is still progressing.

By looking briefly through the New Testament I can think of some examples of individuals' coming to the Lord with a considerable level of contriteness over their sinfulness, for example, the robber on the cross asking Jesus to remember him in His kingdom. On the other hand, there are also some examples of people coming to the Lord for the first time, in which this contriteness is not clearly stated, for example, the eunuch of Ethiopia in Acts 8. 

So, my question is how much realization of one's own sinfulness is needed for a person to be saved?
All the realization of sin in the world can't save someone, they may go and meditate, take up some philanthropic work to make them feel better, kill themselves or find some other way to get away from that realization of the sinfulness of their heart; to try and numb that realization. 

Think of Judas betraying Christ and Peter denying Christ. Judas said "I have betrayed the innocent" (Matthew 27.4) and then you know what he did to himself even though he realized his sin in this circumstance. We know that he was lost (The Lord said before his Crucifixion that he would loose non except Judas - John 17:12) Peter was not lost though! The Lord called him back. Realization of sinfulness doesn't lead us in and of itself to Christ in whom we can be  saved, it is God that does that:


  John 6.44 - No man can come to me, except the Father which hath sent me draw him (call): and I will raise him up at the last day.


I therefore don't think the question is of all that value to think upon in this sense. Maybe a more instructive question would be not "how much" but "whether" realization of sin is a necessary part of the Lord's call and drawing of a person to salvation in the Lord Jesus Christ.