It is the same meaning in both translations. "Bear" in the KJV/NKJV needs to be understood as "carried away", which is the NIV translation. So there is no difference.
The idea is not that the priests have special digestive enzymes that dissolve the sin, nor that the priests have some special virtue that nullifies the sin, but that the priests carry the sin offering into the holy place, because they literally take the meat into the holy place where they sit down to eat it. Moses is so clear about this he repeats himself, so let's read it ourselves:
Lev 10.17b-18 KJV
God hath given it you to bear the iniquity of the congregation, to
make atonement for them before the Lord? Behold, the blood of it
was not brought in within the holy place: ye should indeed have eaten
it in the holy place, as I commanded.
Notice that it is the bearing, and not the eating, that makes atonement.
As it says in Leviticus 6:26:
The priest that offereth it for sin shall eat it: in the holy place
shall it be eaten, in the court of the tabernacle of the congregation.
It is the bringing into the holy place, and really bringing the blood into the holy place -- that atones. To see this, consider the offerings in which the entire animal is burned and nothing is eaten. That happens when blood is sprinkled on the holy place. Nothing is eaten, but the blood is brought into the holy of holies and atonement happens.
Underlying this is the idea that God has limited sight in matters of sin and judgment. This occurs throughout the bible. E.g. Gen 18.20-21:
And the Lord said, Because the cry of Sodom and Gomorrah is great, and
because their sin is very grievous;
I will go down now, and see whether they have done altogether
according to the cry of it, which is come unto me; and if not, I will
know.
The reason God limits his vision is because otherwise it would be impossible for God to dwell among people. Everyone would be instantly vaporized by judgment. So there is an established area of death and of life, with zones of increasing presence: world -> holy land -> the camp -> inside tabernacle courtyard -> holy place -> holy of holies. To make amends, you have to bring the offering into the appropriate level of his presence for the atonement to be made.
But of course sinful people cannot enter that holy place, which is why the priests mediate by interacting with the people and then carrying their offerings into the holy place for them. So yes, the priests are holy, but it is not their holiness which atones, but their holiness allows them to carry the atonement into God's presence.
The eating of the offering is to console the sinner, to confirm to him that his offering was accepted so that his conscience is cleared. It was also a blessing for the priests, that they feed off of mediation. But the atonement happens when the blood is brought before the Lord in the holy place.