The Office of Censorship was a voluntary press code. Obviously "voluntary" in wartime can be less than "voluntary" in peacetime — threatening to denounce as unpatriotic, etc., even occasionally invocations of the Espionage Act as a threat — but it wasn't as heavy-handed, or as monolithic, or as effective, as the name might imply to modern ears. It was also essentially compatible with the First Amendment — there is nothing that says the government can't advise the press on how to behave, or chastise the press if it does things it doesn't like. This is different than actually shutting the press down, jailing its editors, etc. (Whether the Espionage Act is compatible with the First Amendment is a separate question, and it tends to come down to a lot of specific situations.)

Office of Censorship would basically send out bulletins that said, "here are things we really don't want you to talk about, please." If a newspaper or radio show did talk about one of these things, they would work really hard to indicate to the editor/owner that a) they had violated a rule and that was not cool and if they kept it up the government was going to let everyone know who was actually helping the enemies, and b) they would try to make sure that it didn't get syndicated to more stations/papers. Without syndication you could limit the effect to a regional area in those pre-Internet days.

It was also meant to apply specifically to things that pretty much everyone would agree were "war-related." Troop movements and the like. It was not meant to censor political opinions. Obviously there can be fuzzy lines there, but it wasn't as heavy-handed as, say, the use of the Espionage Act in World War I, which was extremely punative to critics of the war effort, in ways that in retrospect almost certainly did violate the First Amendment.

I have looked quantitatively at the effect of the Office of Censorship's diktats against discussion "atomic energy" and "uranium" and things like that. In short, I found there wasn't really any effect on the usage of the term across many national newspapers — there was always a low "hum" of mentioning these terms, both before and after the diktat. There had been a big drop, considerably before the diktat, but that was caused by them basically rounding up all the scientists who worked on these topics and putting them into secret laboratories and stopping them publishing or talking with journalists. So there is perhaps a little lesson there on what actually leads to secrets being kept! There were many leaks regarding atomic energy work during the war, some quite egregious, most pretty subtle. Despite these leaks the Germans and Japanese seem to not have gotten an inkling of the American work.