How did Stalin hide the famines?

With the aid of lots of Professional Journalists.  And the first one who, being an actual reporter, blew the whistle?
The Moscow press corps was even angrier. Of course its members knew that what Jones had reported was true, and a few were looking for ways to tell the same story. Malcolm Muggeridge, at the time the correspondent for the Manchester Guardian, had just smuggled three articles about the famine out of the country via diplomatic bag. The Guardian published them anonymously, with heavy cuts made by editors who disapproved of his critique of the USSR, and, appearing at a moment when the news was dominated by Hitler’s rise to power, they were largely ignored. But the rest of the press corps, dependent on official goodwill, closed ranks against Jones. Lyons meticulously described what happened:

Throwing down Jones was as unpleasant a chore as fell to any of us in years of juggling facts to please dictatorial regimes—but throw him down we did, unanimously and in almost identical formulations of equivocation. Poor Gareth Jones must have been the most surprised human being alive when the facts he so painstakingly garnered from our mouths were snowed under by our denials. … There was much bargaining in a spirit of gentlemanly give-and-take, under the effulgence of Umansky’s gilded smile, before a formal denial was worked out. We admitted enough to soothe our consciences, but in roundabout phrases that damned Jones as a liar. The filthy business having been disposed of, someone ordered vodka and zakuski.

And, of course, politicians who probably didn't give a damn.
...As 1933 turned into 1934 and then 1935, Europeans grew even more worried about Hitler. By the end of 1933, the new Roosevelt administration was actively looking for reasons to ignore any bad news about the Soviet Union. The president’s team had concluded that developments in Germany and the need to limit Japanese expansion meant that it was time, finally, for the United States to open full diplomatic relations with Moscow. Roosevelt’s interest in central planning and in what he thought were the USSR’s great economic successes—the president read Duranty’s reporting carefully—encouraged him to believe that there might be a lucrative commercial relationship too. Eventually a deal was struck. Litvinov arrived in New York to sign it—accompanied by Duranty. At a lavish banquet for the Soviet foreign minister at the Waldorf Astoria, Duranty was introduced to the 1,500 guests. He stood up and bowed.