Is there a way to create an overlapping and integrated dot pattern like this in illustrator without it looking amateurish, like one layer stacked on top of another?
Astute Graphics Phantasm plug in, with its
Halftone option, would make this easier. (Note the plug in is not free, but I believe there's a limited trial or demo available.)
I'm not meaning to imply it will be a simple thing to create the sample image. It will take some work. The plug in merely makes creating the varied size and color halftone-like dots in specific ways easier than what Illustrator has to offer out-of-the-box, so to speak.
This answer is using an older version of the Phantasm plug in (CS6).
Functionality may be better in more recent versions.
Start with a shape that has some sort of gradient to white in it. I chose to use a Gradient Mesh. Then merely call the plug in and adjust settings.
You can then expand the appearance after using the plug in and you are left with "dots" of various colors....
For the particular image in the question. I would create a few different color to white meshes, and then stack them on top of one another, rather than using the multi-colored gradient mesh used for this example. As long as the settings for the plug in remain consistent across different meshes (especially the
DPI setting), the dots should relatively line up across multiple meshes per the sample image.
Disclosure: I have no affiliation with Astute Graphics. I benefit in no way should someone purchases anything from Astute Graphics. However, in the past, I have received free software from Astute in exchange for my opinion.