I've heard many people state that it was the punishment inflicted by the Romans that allows God to forgive us. This just doesn't make any sense to me. How can a man suffer for a short while here on earth in order that a divine being have the right to forgive sin?
Let's start with your question title. For the sake of argument let's turn it around and ask "Why would God not be allowed for forgive us?" The problem lies in God's idea of justice. We could spend a lot of time one this one, but let's keep it it simple:
In order for Him to actually be just, he has to administer justice. In order for justice to be adminstered the proper recompense has to be made for any wrong-doing. We know that the only way to pay for sin is death:
So we have to die. Or somebody has to die for us. The only person qualified to do that is Christ. The why is a subject for another question. Let's skip to what Christ did. When he was on the Cross he took upon himself the full wrath of God against all our sins.
God chose to crush Him. The pain inflicted by the Romans was nothing compared to the weight of that wrath. Even humans can endure the torture of crucifixion for the amount of time Christ did ... longer in fact. The difference was that while the physical act was in progress he also bore an infinite amount of punishment from His own Father. How does that work out for those who believe in Him?
To round this out I would point out that from a Reformed Protestant perspective, the notion that anything the hands of men could inflict upon Christ, no matter how brutal the Romans were could do anything to change God's ability to do or not do something is considered heretical. What Christ endured at the hands of the Romans an in crucifixion is both symbolic of a spiritual reality that was going on behinds the scenes, but also real in that that it put the rulers and authorities of this world to open shame while at the same time setting the stage for his physical resurrection to demonstrate his victory over death.
It's important to remember that there were two separate things happening here. The Crucifixion was only the end of the Atonement, which needed to cover both aspects of the Fall: sin and death.
When Christ was taken by the Roman soldiers, he wasn't still at the scene of the Last Supper, where Judas had been before he left to go betray him. He went up to the Mount of Olives, to the Garden of Gethsemane, and it was there that he took upon himself the sins of the world. He had done everything the Father asked of him thus far, but it was at this one point alone that we ever see any hint of hesitation, when he pleaded, "if it be possible, let this cup pass from me, nevertheless not as I will, but as thou wilt."
Here we have the Perfect Man, who has literally never sinned in his life, being asked to take upon himself the spiritual punishment for all the sins of the world. That's a pretty horrible thing, when you think about it, and it was difficult even for a God to bear. "And his sweat was as it were great drops of blood falling down to the ground." It's hard to imagine what that could possibly be like. Some extreme ordeals, such as childbirth with complications, have been known to cause a very small amount of blood to leak through the skin like sweat. To have it happen generally surely means that something unprecedented is taking place, and to have the subject survive the experience is a miracle in itself!
But he did survive it, paying the price for our sins that we would not have to if we would repent. After that he was taken, betrayed and tried in a sham trial, condemned and sent off to his death on the cross. This was necessary since, even though he had already paid for our sins, there was still death to deal with. So he allowed them to crucify him, giving up his life voluntarily and not having it taken from him (John 10: 18) as evidenced by his sudden death while crying with a loud voice--most victims of crucifixion died slowly of exhaustion, and this would have been impossible ordinarily.
He had to give up his life so that he could take it again, conquering death and unlocking the Resurrection for everyone. It's important to remember that "as in Adam all die, even so in Christ shall all be made alive." (1 Corinthians 15: 22) That part wasn't our fault, so it's paid for unconditionally, for everyone. But our sins are our own choices, so we have to choose to have faith in him, repent of our sins and live according to the law of the Gospel to have them washed away in his blood.
Let us look at Matt 27 verse 27 through 31.
Now, let us look at Isaiah 53 verse 5.
By this explaination, Jesus was pierced for our transgressions, and he was mocked for our iniquities. He was essentially the lamb who was punished instead of us being punished.
Recall that in the Old Testament, God gave instructions to Israel on how to atone for their sins through the sin offering. No longer do we have to sacrifice a goat or a bull, as the price has been paid through Jesus Christ, the Lamb of God.
What granted us God's forgiveness wasn't so much that Jesus endured through pain and suffering, but rather that He offered Himself up for sacrifice.
Jesus provided forgiveness of sins before his crucifixion:
It was not his suffering that cause the forgiveness, but the sacrifice. Had he been beheaded, all the scriptures regarding his sacrifice would be just as valid.
However the way that he died was to fulfill prophecy:
The method of the death itself is not significant (outside of it fulfilling prophecy and allowing him a chance to forgive the thief on the cross). It is the sacrifice he made that allows for the sins to be forgiven