"The people in the west have been considering (separate/separating) themselves from the rest of the country." Which one is correct and why?
Let's ask, "What have the people in the west been considering"?
Since we're looking for "what" (not where, why, etc.), we need a noun to answer that question.
So in this case, we need a noun form of "separate". "Separate" has a noun form--its gerund--which is "separating".
Another noun form of "separate" is "separation". In this case, the sentence would read, "The people in the west have been considering a separation of themselves from the rest of the country." However, the sentence with "...separating themselves.." sounds more natural.