Jesus was able to pray to the Father directly because, as the actual Son of God, he did not require anyone to intercede on his behalf or act as a mediator between him and God.
That is why it so amazing that Jesus told his disciples they too could address God as 'Father'. It's all about relationship with God. Christians are adopted as sons/daughters and are co-heirs to the promise. Check out Romans chapter 8, but especially verses 14 - 17: "because those who are led by the Spirit of God are sons of God. For you did not receive a spirit that makes you a slave again to fear, but you received the Spirit of sonship. And by him we cry, 'Abba, Father.' The Spirit himself testifies with our spirit that we are God's children, heirs - heirs of God and co-heirs with Christ, if indeed we share in his sufferings in order that we may also share in his glory."
See also Galatians 3:12 and 4:7; Ephesians 3:6; Hebrews chapter 1 and 1 Peter 3:7.
What a glorious privilege, to enter into relationship with our heavenly Father because we are now adopted! But Jesus, as a literal Son of God, was able to speak directly to his heavenly Father.
With regard to what Jehovah's Witnesses believe, they say only 144,000 people since the time of Jesus can be in that relationship - be anointed and be heirs to the promise.
However, Jesus said (in John 14:13-14) that anything we ask for in the name of Jesus, he will grant - the literal translation of John 14:14 is 'If ever anything you should ask me in the name of me this I shall do.' That is why it is fitting and proper to pray to the Father in the name of the Son. After all, they are one and the same!