The healings of the two blind men as recorded in Matthew 20:29-33 are not the same men that Jesus healed in Mark 10 or Luke 18. The Matthew 20 account tells of two men together, both of whom received their sight and followed Jesus.
Luke 18:35-43 records what happened while Jesus was on His way to the city of Jericho, where there was a blind man sitting by the road begging. He heard the multitude of people passing by and he asked what was going on. He was told that it was Jesus passing by. He immediately shouted, “Jesus, Son of David, have mercy on me!” He was warned that he should be quiet, but he knew this might be his only chance. He shouted even louder, “Son of David, have mercy on me!” Jesus stopped and asked that the man be brought to Him. Jesus then asked the man, “What do you want Me to do for you?” He said, “Lord, that I may receive my sight.” Jesus said, “Receive your sight; your faith has made you well.” Immediately he received his sight and followed Jesus, glorifying God. The crowd of people also, when they saw what had happened, praised God.
The next healing miracle to be considered is in Mark 10:46-52, which begins with “Jesus and His disciples came to Jericho.” However, no details are given of what occurred in Jericho. Only Luke 19 tells of what Jesus did while in Jericho, most memorable for His meeting with Zacchaeus. Mark 10:46 continues with Jesus leaving the city with His disciples and a great multitude following Him. A blind man named Bartimaeus sat by the road begging. When he heard that it was Jesus of Nazareth, he, too, began to cry out, “Jesus, Son of David, have mercy on me!” Similarly, he was told to be quiet but only shouted louder. Jesus stopped and commanded him to be called. Then they called the blind man and said to him, “Be of good cheer. Rise, He is calling you.” He got up and came to Jesus. Jesus asked him, “What do you want Me to do for you?” Bartimaeus said, “Rabboni [my master], that I may receive my sight.” Jesus said, “Go your way; your faith has made you well.” Bartimaeus was immediately healed and also followed Jesus on His way for the last time to Jerusalem.
Jesus always required believing on the part of those receiving healing. The only exceptions were dead people, demented people or small children incapable of faith. “Do you believe that I am able to do this?” (Matthew 9:28) “Do not be afraid; only believe.” (Mark 5:36) “Therefore I say to you, whatever things you ask when you pray, believe that you receive them, and you will have them.” (Mark 11:24) When Jesus was in His home town of Nazareth, “He did not do many mighty works there because of their unbelief.” (Matthew 13:58) Whenever someone was capable, Jesus demanded faith, and let them know in no uncertain terms that it was their faith that brought to them their desired results. “O woman, great is your faith! Let it be to you as you desire.” (Matthew 15:18) “Your faith has saved you. Go in peace.” (Luke 7:50)
Often Jesus asked what might seem to be obvious questions. “What do you want Me to do for you?” Jesus could see that these men were blind, but He knew that they had to specifically articulate their requests in order to focus their faith on exactly what they were believing for. This is still true, and although God already knows every need and desire of His people, He instructs believers to be specific in their requests. “…let your requests be made known to God.” (Philippians 4:6) Jesus asked these men to state their desire. “Lord, that I may receive my sight.” There was no need to lay hands on them, or make a dramatic scene. Jesus knew inside spiritually that they had the faith to receive their deliverance and He told them so. “Receive your sight; your faith has made you well.”