It's for a story I was writing, and in the story, there's a section of the newspaper in which people can submit stories/poems/essays to, and good ones were chosen to get published on the newspaper. These people could get money for getting their work chosen and published. But I don't know how much they should get (how to calculate it? Like for example, 10 cents per word?), and do they just get paid, or do they need to sign a contract or something?
AFAIK, people usually don't get paid for that sort of thing. It's like a letter to the editor. The reward is getting your story/poem/essay in the paper. The small amount of money that somebody would get paid for a one-time short submission isn't worth the newspaper's bother.
But, if it's a weekly contest, then the newspaper might want to increase interest by having a small monetary reward also. However, whatever amount you choose could easily sound silly in 10 years, and would probably not translate well to other countries. So, I'd suggest that your story's newspaper offer something to the winners like a one-month subscription, or a basket of groceries (sponsored by the page's advertiser, a local grocery store).
Note: This question seems familiar.
Freelancers (writers who aren't employees but get paid for writing assigned articles) get paid differently than contest winners, when a publication has solicited entries from the public.
Freelancers can get $1 or more a word from large publications for articles of 500 to 2,500 words, as a rough example; I've been paid that for articles requiring a week or more of reporting and writing.
Contest-type awards are usually less. In today's media market, a publication might pay a nominal amount -- $100, maybe $250 -- for things like poems and short essays. A big, national publication might pay more to winners of a high-profile contest, like $1 or $5 or $10 thousand even. But that would be for a big contest -- and very good writing.
The award for getting something published could also be nothing, or a token prize, like a framed copy of the article, a 1-year subscription or a gift card for $25. The pay would essentially be whatever the publication thinks it needs to offer to get enough good submissions.