Hello. Hopefully this question isn't too vague but in the example I included with Putin's face, would anyone have any insight as to what filter they are using or how the grain effect was achieved?
This is one of those infinite ways to add this effect situations.
Camera RAW > Effects > Grain. This used to work much better in earlier versions as you could control the size of the grain as well as intensity. Nowadays it's just the intensity.
Filter > Texture > Grain > Enlarged. intensity was set about 50%
This is more by way of a demonstration. If you place a layer above the layer you want to effect and fill it with 50% gray and set its blend mode to Overlay it disappears. However, and effect you apply to the 50% grey will be applied to layer below. I used Filter > Sketch > Reticulation here. I have left an outline of what it looks like with blend mode set to normal.
Because there are little flecks in the film grain, an easy way to replicate the effect would be to get a film grain image (something like this one from Adobe Stock), add it as a layer over your image, and apply a Blend Mode such as Screen. You could try other blend modes to see what looks the best.