I've heard both usages in the meaning "to get caught", however, I'm not sure that both of them are correct.
From my research, though, both should be fine, however, "got" is both Br.E and Am.E, while "gotten" is only Am.E, because:
In American English, the past participle of “get” in its literal sense of “receive” or “become” is usually “gotten”.
There another interesting thing about "get got" vs "get gotten". That is, when we use them in the Past Perfect tense the "gotten" version is preferred due to the doubling of "got". I can't say for sure about the Present Perfect or the Future Perfect since I've never met either of them used in these tenses.
- He had got gotten (not got got) yesterday by the police.