Student or Learner
1) This issue will become a news on the newspaper that everybody will talk about.
2) It will become an international news which everybody will talk about.
1) This issue will become a news story in the papers, and everyone will talk about it.
2) It will become a piece of international news and everyone will talk about it.
The problem with "that/which everyone will talk about" is that in both your sentences that part appears to refer back to the most recent noun. In sentence 1 it appears that everybody will talk about the newspaper, and in Sentence 2, that everybody will talk about the international news.
It's also difficult to predict that something will definitely be a story in the papers or in the international press generally, because the news changes, minute by minute. What looks like a big story now might be superceded by something far more important by the time the papers come out. Is there some specific reason why you chose to write your sentences in the future tense?