Im currently working on an art collage project where I want to cause the dyes and inks from magazine cutouts to bleed, smear, and run across their paper. I have tried paint thinner, bleach, vinegar, fire, but nothing is creating this bleed effect that I desire. Is there some household products I can combine to create this effect? Are there chemicals that I can buy that would be relatively safe to use (indoor or outdoor)?
A technique that will get the result you are looking for would be to transfer the imagery from the magazines to something that you could then transfer again onto your work of art.
Color copies of the 'zines would probably work best, saw a lot of that in art school. Not sure the solvent used on the copies but pretty sure it is one of the things you tried for the magazine prints, probably rubbing alcohol. That will give the running and smearing effect that a "proper" transfer would lack.
The magazines themselves are printed in inks that are too permanent to dissolve before the paper breaks down.
The copying process allows you another point of manipulation as well, as you can scale the pictures up or down, copy multiple images, move them around during the scan, etc...