Other than a material safety data sheet (MSDS) or a completely outfitted laboratory and mad science skills, there is generally no way to know for sure what chemicals are actively producing color effects or passively integrated as fillers / extenders / driers. Technically, stores are required to give out manufacturers MSDS upon request. If they don’t, you can report them to the local business bureau. You can also contact the manufacturer; they are required by law to send you MSDS information free of charge.
A good reference work about the general composition of paints is a recent edition of The Artist’s Handbook. There you will find pretty much bleeding edge information about the state of the art in chemical hues etc.
One thing that is rarely talked about however, is the fact that modern pigments - even technically inert ones like titanium dioxide - undergo a crushing process that creates nano-particle sized pigment components. As is well known, nanoparticles can easily traverse cell walls, and can lead to DNA mutations or simply cell death. This is why I highly recommend never getting any paint on your skin - especially those made from heavy metals like cadmium and lead.