How do I achieve an “old” distressed poster effect in Illustrator, Photoshop or Pixelmator?

There are many ways to do this and to achieve this effect very easy you can just use a ready-made texture (search for example for “vintage” or “letterpress”).

If I wanted to make my own texture I would use photoshop and start with a new black layer on top of my image.

Then i would add a low amount of Noise to it (Filter > Noise > Add Noise …)

Now press cmd/ctrlF a few times to add new instances of the effect on top and make the noise overlap in some places. This way you will get some dots that are brighter than others and therefore more variance in the noise. You can adjust this effect by using Curves or Brightness/Contrast.

Now add a Layermaskand select it.

Use Filter > Render > Cloudson the Layermask (Clouds will use your selected fore- and background color, so don’t forget to reset them to black and white by pressing D). This will make the noise visible in random patches.
Now you just have to set the blend mode to Screen.

I bet if you play around a little with the settings and maybe also try out some of the airbrush brushes in Photoshop, you can get a good result.