This is quite impressive. I almost studied photography collegiality, so I spent a ton of time in darkrooms growing up. There's a couple of things that they could have done in order for him to be removed from the final print. Darkroom print editing is always tricky and usually difficult to do.

Option 1) The negative from the film itself was edited, cut, and reassembled prior to light exposure. They would have had to razor-blade Nikolai Yezhov out of the negative and replaced it with a snippet from another film negative. This is incredibly difficult to pull off. Depending on the size and type of film, this would prove to be quite meticulous. He could have been cut out, film exposed with a blank silhouette of Yezhov, then another exposure from the site used to fill in the blank.

Option 2) The light could have also been artificially exposed onto the paper by blocking certain spots with strips of paper or other material to give the appearance of a complete photograph. These are called "photograms". On the darkroom lamp, the tech would placed paper strips, string, or any combination of necessary materials onto the paper. These materials block the light and can provide the necessary shape to totally fake a photograph. Through trial and error, the photo could have definitely been completed by a skilled darkroom tech without taking another exposure. This is the best strategy IMO.

Option 3) The paper, under the lamp in the darkroom, was screened in the section of the photo where Yezhov stood. After one exposure, Stalin's half of the paper would be then covered. The second half of the photograph could be completed with another film negative from the same spot taken afterwards. It wouldn't necessarily have to be in the ideal angle when the second photograph was taken, it's fairly easy to position the lines of the ledge where they would line up with the ledge in the previous image. I don't know anything about the equipment used at any step of this process, but early film was incredibly versatile. This option seems unlikely to me, the edge of Stalin's sleeve is too clean.

So, cutting Yezhov out of the film negative or screening Yezhov's space on the paper under the lamp is most likely. After the tech had a blank Yezhov-shaped space to fill, every other inch of the paper was then covered for lamp exposure 2. That space was then most likely filled with another negative of a naked ledge OR by placing objects to darken the proper areas. The exposure time and aperture were most likely available. Black and white film (as well as printing) is conveniently the easiest to fake as well. The tech just had the task of keeping lines straight, matching saturation, and replicating and completing the patterns of the water.

It probably took a very long time to make it look this damn good.

During the late 1930s, if Stalin and the Communist Party decided you were a "former Person" they would not only kill you, they would remove any trace of your existence. There is a book about how this was accomplished: "The Commissar Vanishes: The Falsification of Photographs and Art in Stalinist Russia." by David King (1997) It is often said that they just air brushed Yezhov out of the picture, but that is a crude method that results in obviously faked photographs. To make it look good, requires a time consuming process, that is detailed in this book. /u/petercartwright provides a good TL/DR summary. In this particular photo, the Soviets used the third method /u/petercartwright mentioned.