A lot of people worry about copyrighting their stories ahead of time, to prevent unscrupulous publishers from stealing their work. This is generally a needless worry—at least where submission to reputable publishers is concerned.

Copyright

The U.S. Copyright Law provides you with copyright protection from the moment you set the words down. If and when the story is published, the publisher should provide the appropriate copyright notice and register the work with the U.S. Copyright Office. After publication of a short piece—for example, a short story or other contribution to a magazine or anthology—it's a good idea to register your contribution separately with the Copyright Office. (For book publication, the publisher's registration should take care of that.)

I have never heard of anyone having their work stolen or plagiarized as a result of submitting to a recognized publisher.

Plagiarism Happens

Of course it does, but usually after something has been published. The internet has made it easier to do—but also easier to detect. My own work was stolen once after it was published in an SF magazine. But that wasn't the fault of the publisher. When the theft was discovered, it was easy to prove. (The student who did it was ultimately expelled from college. See Plagiarism—Don't Do It!)

(Hollywood is another story, but that's beyond our scope here.)

Copyright Notice

It doesn't hurt to put your own copyright notice on the manuscript, at the bottom of the first page:

copyright © [current year] by Joe Q. Author

but it's not necessary, either.