In this passage we need the Greek text in order to make an accurate interpretation. First, the pronoun αυτός, which means, basically, "him," is used three times. Let me highlight where it appears:
19 My brethren, if any among you strays from the truth and one turns him back,
20 let him know that he who turns a sinner from the error of his way will save his soul from death and will cover a multitude of sins. (ESV)
The author of the text is not out to confuse his readers, and as a Greek reader one can therefore be sure that each time the pronoun is used it refers to the same person. The first "him" translated in verse 20 is very confusing and it surprises me that the ESV team translated it thusly. The pronoun used is τίς, as opposed to αυτός, and it means, more accurately "certain man."
Here's my translation from the Greek:
My brothers, if ever a certain one among you should error, away from the truth, and a certain one should turn him, know that the one who turned the sinner from the error of his way will save the soul of him out of death and will cover many sins.