Albert Barnes Commentary Numbers 19:3

"And ye shall give her unto Eleazar the priest, and he shall bring her forth without the camp, and one shall slay her before his face:" — Numbers 19:3 (ASV)

The work would necessarily require a priest; yet, as it made him unclean for the day (Numbers 19:22), the high priest was relieved from performing it.

Without the camp - The defilement was viewed as transferred to the victim that was to be offered for its removal. Under these circumstances, the victim, like the defiled persons themselves, would be removed outside the camp. The particular pollution to be remedied by this ordinance was the indirect one resulting from contact with tokens and manifestations of sin, not the direct and personal one arising from actual commission of sin.

Likewise, the sinless antitype had to bear the reproach of associating with sinners (Luke 5:30; Luke 15:2). And as the red heifer was expelled from the precincts of the camp, so was the Saviour significantly cut off during His Life from the fellowship of the chief representatives of the theocracy, and put to death outside Jerusalem between two thieves .