"Rights afterwards I do not see him anymore."
It also doesn't track. "Rights afterwards" makes no sense--rights are liberties; I don't think any other sense of "right" is ever plural. If they mean "right afterwards"--that is, "immediately afterwards," a short defined period, it doesn't go with "anymore"--an infinite period, forever.
At first, this example really lacks some past tense. But I'm actually puzzled by your last remark. I imagined the speaker to be interviewed by the police about someone: Right afterwards (the alleged crime, for ex.) I didn't see him anymore (like: no more, not again until now). How would you actually say that in English?
I would probably say "Afterward [OR after that], I didn't see him anymore."
The problem is in the phrase "RIGHT afterward[s]"; the "right" contains the action [the not-seeing-him] to the time immediately following the crime. For example, in the same situation, I might say "Right after that, he disappeared."
A similar usage would be in reference to place. "He was right there" emphasizes that he was on top of the scene or practically so--not somewhere nearby, not at a distance looking through binoculars (except perhaps relative to being in a remote country).
Also, In formal writing, "afterward" would usually be preferred to "afterwards," but "afterwards" is not uncommon in spoken or informal contexts.
Thanks for the explanation. The german 'unmittelbar' has the same weight as 'right', it's very immediate. Now that you named it, in this example, danach or anschließend (= afterward) alone might actually fit better, since 'unmittelbar' follows the same logic as your explanation.