If your image is a screenshot from the Illustrator workspace, chances are very good that this is simply a preview artifact, and not something that will show in your exported files. But there are a couple of caveats to this, based on your statement that "it gets worse when you save to PDF". But first, let's take care of your preview problem:
First, I created a quick illustration with grey layers and subtracted the shapes with the Pathfinder, similar to yours, and put a bright magenta background behind it. In Pixel Preview mode, this is what I see (at 300% to minimize the imgur lossiness effect):
So, clearly a reproduction of the sort of problem you're having.
However, without any modification, when I Save for Web as PDF, here's what the output looks like (again, 300%):
Perfect anti-aliasing between the shapes, with no magenta.
While Pixel Preview is great for fine-tuning before export, even for screen/pixel design, I tend to do most of my work in GPU Preview mode, so I'm not needlessly distracted by jagged edges and bleed effects. I only switch to Pixel Preview when I need to do detailed alignment or similar tasks. For a real pixel preview, I always check the exported file in Photoshop for anything even remotely important. The only catch is, you need a compatible graphics card for GPU preview mode to work.
Thus, the bleed-through effect you're seeing in Illustrator should go away in your saved files, and can be "turned off" within Illustrator in GPU preview mode.
What if it doesn't go away?
You need to revisit your illustration and look at how you subtracted your shapes. If you resized, shifted, filtered, or otherwise did anything that might alter the outline of your shapes, that could create a persistent version of the problem. You can quickly check this by going into Outline Mode, zooming in to 8000% or so, and verifying that your shape boundaries line up. They should appear as one line at any zoom setting.
Your PDF problem (why raster?)
You've said that the problem is "worse" in your PDF. However you've also said that you're saving these files as PNGs, with other images behind them. Without knowing more about that exact process, I have concerns:
Adobe PDFs support vector art. In fact, that's one of the format's strengths. Why are you exporting to PNG (a raster format) before bringing that in to your PDF design? It's quite possible that the bleed-through you're seeing there is a result of any number of raster alpha channel/zoom/display effects. While the other answers might help you mask the problem ...
is there a reason you can't keep your illustration in vector format?
Even if your design is going to overlay something that must be rasterized (like a photograph), there's no reason I can think of why you can't keep your scribbles in a vector format. If you still see artifacts, then that would be another question, but I've done many similar designs that have displayed perfectly at any zoom level in about a dozen different PDF viewers that I test with.
Digital problem in general (if PDF isn't your actual destination format)
If the information I've given you here doesn't help, then please let me know more about your actual destination format and your export/merge process with the other images that you alluded to.
In general, you should never have to grow outlines assuming a simple Pathfinder slice job with Illustrator (in fact, that can create other problems), and no shape-altering modifications after that. The only time I've had problems is with quick jobs where I had to go through lossy formats, multiple exports, and that sort of thing. Keep your workflow as simple as you can, and make sure you understand where and how your vectors are being rasterized—that step tends to define the success or failure of such effects.