The first photographic images printed in newspapers were actually wood engravings meticulously hand-copied from a photograph printed in the normal way. By the 1890s, however, prints were made in essentially the same way they are today: through halftoning — printing different tones as patterns of small dots varied in size and spacing. By the 1929s, this technique was relatively sophisticated, although arguably the image quality afforded by hand-engraving was still much higher, but hand-engraving also required considerable artistry and time (and therefore expense). More sophisticated ink prints could be made through photogravure, and while those were used for high-quality books, that process was also far too expensive for newspapers, advertising flyers, or cheap magazines.

Halftones were made like this: the original printed photograph was re-photographed through a glass screen with a pattern of tiny apertures, onto a film or a plate. This was then developed at very high contrast, resulting in dots which varied in size according to the intensity in the original. This, in turn, was used to make a sort of contact print on a sheet of metal using a material which would harden when exposed to light. The rest of that material was then washed away, and acid etch used to dissolve the bare areas between the dots. This resulted in a plate which was used in the printing press. (It'd be fastened to a wood block and locked into place along with the type on the page.)

If you have a higher-resolution image, or the original, look closely (zoom in, or use a magnifying glass) and the halftone dots should be readily apparent.

Here's a crop from a postcard from around 1910, clearly using a very simple single-screen halftone process:

(This image is grayscale, and wasn't scanned at high enough resolution to make it possible to convert back to pure black and white without losing detail, but if you were able to look closely at the original, you'd see that there's obviously no different tones of ink — just the black.)

A more sophisticated method involved doing this multiple times with screens of different sizes, but that didn't become common until the 1930s — after your stated time. By the 1970s, this technique was basically replaced with photo offset printing (where the whole page, text and all, is transferred to the plate photographically), and of course CMYK printing added color, but the fundamental approach of halftoning remained — and does today, although the screen is now almost always digital.

If you're interested in this kind of thing — and the development of photography as a physical art overall — I highly recommend The Printed Picture, by Richard Benson, which covers this and many other print processes from the Renaissance to now. If you're looking for more on newspaper printing in specific, searching for more about halftones, letterpress, and offset printing should get you in the right direction. Encyclopedia Britannica (still a thing!) has a nice aricle on photoengraving which covers this process as well.