|For the ears, not the garden...|
[Do you compost] paper with ink on it?TL;DR:
Before I started, I looked long and hard at that, because I had read that inks weren't safe, newspaper was bleached, all that fun stuff. But while everyone was saying it, very few people actually linked to a source. It was just one of those things that everyone knew. Or at least said.
Having worked in a print shop, I was suspicious of the claim that modern inks are dripping with heavy metals just waiting to kill you. The vast majority of inks today are made from soy, and the 'glossy' paper is generally covered with kaolin clay, not plastic. Kaolin is inert and will break down easily if your paper is shredded.
BPA in the ground has a half life measured in hours: An important result of the degradation study was that, independent of the soil type, 14C-BPA was rapidly dissipated and not detectable in soil extracts following 3 days of incubation. So shredding that Walmart receipt is probably ok.
As one of the few articles that actually uses sources noted about PCBs: Since all carbon sources take up PCB 11 from the air, and it is present everywhere, that really leaves us with no uncontaminated source of carbon-rich material. Like cadmium, PCBs can be found in inks in small traces, but because it's in the air, it's found in small traces in everything, including your garden plants, which take it up through the air much more easily than they do through the roots*.
Is there risk that something bad for you is going to get into your shred through inks? Absolutely.
But as there's no way to completely avoid pollutants, it's really a matter of risk management. I look at it this way: if you don't worry about your kid chewing on a magazine page, there's no need to worry that he'll be harmed when that same page is shredded, rotted, run through a worm or roly poly**, mixed in with a ton of other things, then used by a plant that then goes into his mouth.
If you don't mind that the melted cheese from your pizza touches the box, why would you mind if the box ends up in the garden?
I strip out plastic windows, plastic tape, obvious metal inks (magnetic and fluorescent), but otherwise I don't worry about it too much.
UPDATE: On a completely related subject -- and going the other way -- is the prospect of what David the Good calls killer hay. There are herbicides used in agriculture that even once they pass through plants, then animals, then compost, will still kill your garden.
That stuff is probably in your super-meat pizza: in the sausage, the cheese, the tomato-based sauce, and the dough. You might just be safer eating the box.
* So if you want to keep PCBs out of your garden, bury the newspaper paper instead of burning it.
** Which bugs remove heavy metals, like cadmium, from the soil.