When the news media makes a significant fact error in a story, it is perfectly all right — and, in fact, a good idea — to ask for a correction.
The same advice doesn't apply when you simply dislike "the other side of the story" contained in the coverage.
There is no magic in asking for a correction. Start by calling the reporter who wrote the story. Most times, they are eager to clear up any mistakes. Before running a correction, they will (and should) verify your claim there is a mistake. You can help by providing a credible or official source of the correct information.
Some people wonder if a correction is worth the bother. Others fret it simply brings additional attention to a story you would just as soon have fade away. Those concerns are misplaced.
You need to stick up for your facts — whether it is the correct spelling of a name or an accurate description of a legal process. Reporters and editors don't resent that; they respect it. A constructive, polite exchange about a correction can actually establish better rapport with reporters and editors, paying dividends in future coverage.
Errors in print can be frustrating. Corrections appear in later editions, often in a section reserved for corrections, not necessarily with the same page dominance as the original story. However, most archival searches nowadays occur digitally, so a correction for online editions can be worth the effort. If you rigorously monitor a story's appearance online and spot an error early enough, you sometimes can avoid the mistake appearing in print, at least in later editions.
TV and radio newscasts seldom run corrections, except for the most egregious errors. However, they also have websites where corrections can be made so errors aren't perpetuated. Some broadcast shows, such as NPR's All Things Considered, have a section devoted to reader letters, which often point out mistakes or poor news judgment.
Bloggers may hang out somewhere between credible journalists and eager hobbyists, but they also should be given attention when they make a significant fact error. A student in one of my classes who operates a discount website was upset when a blogger essentially re-posted an old piece about the site's deficiencies under its prior ownership, I encouraged him to call the blogger to remind him of the change in ownership and the steps taken to address the faults he identified. The call was an opportunity to get a positive post, contrasting the new with the old.