Mafia 73. Forest Kingdom

"On himself" meaning on Satyr, or "on himself" meaning himself?

On Satyr. Like a skeleton in the Pliskin mafia.

I'm more interested in what the exchange is: the Satyr gets the player's negative effect, and the player gets his? Is that how it works?

Yes. Let's say player A goes to player B with a specific negative effect, and the Satyr exchanges his negative effect for it, and player A takes on the negative effect of player B. This could lead to some interesting situations, so it can be quite funny. If player A was going to kill someone, the assassination is thwarted, but the Satyr himself won't be able to apply the assassination the next night. He won't know the negative effect.