Is Every Christian Supposed To Win Souls?

I believe so, yes. Part of The Great Commission is to "teach others" the things which Christ taught us. The idea is to perpetuate the Gospel...

Matthew 28:18-20, “And Jesus came and spake unto them, saying, All power is given unto me in heaven and in earth. Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost: Teaching them to observe all things whatsoever I have commanded you: and, lo, I am with you alway, even unto the end of the world. Amen.”

So we see as believers we're supposed to:

1. Go into all the world
2. Preach the Gospel
3. Water baptize converts
4. Teach them all things concerning Christ; and then they hopefully repeat the cycle, which is the whole idea

CCM singer Amy Grant is infamous for this horrible remark:

“I'm a singer, not a preacher, I'm not looking to convert anybody” says Christian rock diva Amy Grant. (Los Angeles Times, 5/4/84, pg. 2-c) Grant goes on to demonstrate her spirituality by saying, “I'm not going to say too often that I like a cold beer while watching a football game. That might bother some of my fans.” (Greensville News, 5/4/94)

I won't judge her for the beer comment (although I think as a role model to youth she shouldn't have said that), but she is wrong to say that she's not trying to convert anybody. It just shows how carnally-minded and spiritually barren the Contemporary Christian Music (CCM) industry is. K-Love is the most popular CCM radio station nationwide, who rake in over one billion dollars each year, but they are fruitless. They propagate a ridiculously shallow religious message of making a commitment to Christ, which is not the Gospel. Daniel 12:3, "And they that be wise shall shine as the brightness of the firmament; and they that turn many to righteousness as the stars for ever and ever." Proverbs 11:30, "The fruit of the righteous is a tree of life; and he that winneth souls is wise." 

The parable of 'The Sower And The Seeds' in Luke 8:11-15 it is about soulwinning, not salvation. Luke 8:15, "But that on the good ground are they, which in an honest and good heart, having heard the word, keep it, and bring forth fruit with patience." Kindly, Evangelist Ray Comfort gets this passage way wrong in his 'The Evidence Bible,' teaching that the second group in Luke 8:13 got saved, but then lost it. They did get saved, but they didn't lose salvation; the Bible says they had shallow roots and fell away, not maturing to BRING FORTH FRUIT. God's goal for every believer is to mature so that they will also bear fruit. This passage is all about maturing to bear fruit, evidenced by verse 15, which says an obedient believer brings forth fruit, which takes time. 

The Bible says we bear fruit "with patience." I have received many e-mails from people over the past 18 years, who said they found my website, and thought it was either a joke or I was nuts, but then a few years later (as the truth worked on them) they came to Christ for salvation, and now they love me. I love testimonies like that! Many people won't get saved the first time they are witnessed to. That is why the Apostle Paul said in 1st Corinthians 3:6-9 that some Christians SOW a seed, and then another Christian WATERS that seed, and eventually, still another Christian REAPS the soul saved; but only God giveth the increase (i.e., only God can save a person). So our responsibility is to witness the Gospel to everyone (to "EVERY CREATURE," Mark 16:15). That includes the vilest of sinners and criminals, who are "creatures" too. Thus, I believe Biblically that soulwinning is every believer's duty to Christ. 

We read in Luke 2:36-38 about an 84 year old woman named Anna, who watched for people travelling to Jerusalem to find out about this man named "Jesus" whom they had heard about. She prayed and fasted, and shared the Good News of Christ crucified, buried and risen with visitors seeking answers, leading them to the Lord. Bless God, if an 84 year old woman can serve the risen Savior, then we all can. I love this truth in the Bible.

“'Pray to be saved. Ask the Lord to save you.' This is what I heard a minister say to a convicted sinner who came forward for prayer at the close of an evangelistic service. Nowhere in the Bible are men told to pray to be saved. There are examples where men did pray and were saved. Men are told to repent, to believe, to obey the Gospel, but the Bible never says to pray for salvation. It does say that whosoever shall call upon the name of the Lord shall be saved, but nowhere does the Bible command men to pray to be saved. I think I know why, God, who made the human heart, knew that it was not necessary to tell convicted sinners to pray. If a sinner can see Jesus Christ on the cross, it is as natural for him to pray as it is for birds to sing or flowers to bloom. 'Believe on the Lord Jesus Christ and thou shalt be saved.'” —Dr. Bob Jones Sr., “Comments On Here And Hereafter,” chapter: Prayer And Praise, page 163; Bob Jones University Press, Inc., © 1942