Yes, this was a big reason why it was initially named the Spanish Flu. At the time that it began (with the war still on) Spain was one of the few countries that did not have very stringent press censorship in place.
This is discussed in several sources, but this is the one I had most readily available:
"In a month or two everyone outside of Spain was calling it "Spanish influenza," not because it originated there, but probably because Spain, still a nonbelligerent, had no wartime censorship to keep its health problems secret from the world. An estimated eight million Spaniards caught flu in May and June. The Spanish claimed that it had come from the battlefields of France, blown over by the strong winter winds, and that it would have been even worse but for the snowy Pyrenees."
America's Forgotten Pandemic: The Influenza of 1918 by Alfred W. Crosby
I don't have a ready source, but the press in the US would usually comply with a request to censor news that some in government worried would lead to panic, blame, or accountability. Continued well into the 60's and 70's, and still present today.
But IIRC there were dramatically more strict restrictions during WWI. Certainly in all the european countries busy fighting. (Famously, Bertrand Russell got himself locked up for six months for giving a lecture about whether the US should enter the war.)
This is discussed in several sources, but this is the one I had most readily available:
"In a month or two everyone outside of Spain was calling it "Spanish influenza," not because it originated there, but probably because Spain, still a nonbelligerent, had no wartime censorship to keep its health problems secret from the world. An estimated eight million Spaniards caught flu in May and June. The Spanish claimed that it had come from the battlefields of France, blown over by the strong winter winds, and that it would have been even worse but for the snowy Pyrenees."
America's Forgotten Pandemic: The Influenza of 1918 by Alfred W. Crosby