Most newspapers still run free obituaries as well as paid death notices. But the free versions are generally limited in the types of information contain, and as newspapers have cut back on their size they've also reduced the amount of detail the obits contain. At the newspaper where I work, it's pretty much name, age, where the person lived and when and where the funeral is. (Some of the really big papers have eliminated the free ones altogether).
The paid death notices, on the other hand, function more like classifieds: You can write it any way you want, and you pay by the line or the word.
What most people don't realize, though, is that the funeral homes charge a hefty free even for providing the newspaper with the barebones information for a free obit. And some funeral homes do an appalling job. One major home in my city seems to pick the employee with the most God-awful penmanship to scrawl the information by hand; names are hard to read and details like place names are often misspelled. One of my first newsroom jobs largely consisted of fact-checking the info the funeral homes frequently got wrong (incorrectly putting two t's in "Paterson, NJ," stuff like that). Of course, the papers have pretty much cut all those jobs as well.