When we pray, does Jesus hear us? Is there a direct point of contact or is it received by all three persons of the trinity simultaneously?
There is no way to know for certain, and given that God knows your prayers before you pray them, if you will, before the foundation of the world, the direct path of your prayer is somewhat immaterial.
Those who dwell in him, indeed, must also have this knowledge as well; but in a more limited fashion since they are not God.
For the second part, the Father, Son and Holy Spirit fully indwell one another, thus aside from the aspect of incarnation, it is impossible to make such a distinction.
Short answer: My understanding is that "we pray to the Father, in the name of the Son, with the help of the Spirit."
Let's say you write a letter to the CEO of a company: one important thing you need is the address. Even if you take it there in person, you first meet the front desk manager/assistant or receptionist. The letter goes through him or her before it is handed over to the boss.
Jesus is the address or the receptionist, the path.
The Holy Spirit is God's active force that we pray for, for him to use to empower us and direct us to carry out his works.
Well don't try to imagine that there is Complicated Prayer System or algorithm in heaven.
Just hold this very word of Jesus:
The prayer path is simple one: ask and you'll receive the answer.
Please read Luke 11:1-13 so we may understand what was exactly Jesus trying to teach us regarding prayer. Let us beware so that our intelligent didn't hinder us from understanding God.
In response of @waxeagle 's demand:
First, let us see what's Jesus intention:
The possible question arises in the people's mind is: What's the difference between this kingdom of God with all the world's kingdom so that it's important to be preached?
Jesus provide the answer through Lord's Prayer as we already know it:
Unlike all the world's kingdom, God, the King himself is our Father. The hassles (path, system, rules, algorithm, laws) we going through to speak (pray) to the King is present no more. I believe everyone here can easily talk to their own father.
Examples some of the hassles before the time the kingdom of God is preached:
In the evening in Galilee.
Disciples: Rabbi, how's the rules (path, system, algorithm, laws) so that our prayer reach God (King)?
Jesus: It's simple, there's no rule, just go and talk to Him. He's our Father after all.
I hope it helps. God bless you.