That depends on the type of paint mixed in.
Polymer clay is oil
-based (not water-based like air-dry clays are), so the best paint to mix into them as a colorant/etc would be artists' oil paints (you can buy them in small kits of tubes in different colors to save money, and can mix the paints to create new colors before mixing them into the raw clay if desired---or mix colored clays together).
Other possibilities would be shavings from oil pastels and various dry pigments like mica powders (Pearl Ex, etc), other art pigments, even spices like paprika/turmeric. Other larger-particle colorants can also be mixed into polymer clay like colored sands, herbs, glitters, etc.
Water-based colorants like acrylic paint can sometimes
be used without adverse effects but mostly only if a fairly small amount is used, and the clay is allowed to sit for awhile before baking (to allow some of the water to evaporate out).
You can get more info on all those things if you're still interested on these pages of my polymer clay "encyclopedia" site:http://glassattic.com/polymer/color.htmhttp://glassattic.com/polymer/inclusions.htm
On the other hand, water-based paints like acrylics are usually used on top of
cured polymer clay.
Oil paints could be used too but would take a long time to dry.
It's the paints (and clear finishes) that are petroleum
solvent based that are not compatible with cured polymer clay since they will begin eating into it immediately or after some time (those would be paints that must
be thinned or cleaned up with paint thinner, etc).
(If you have more questions about polymer clay, ask them in the Polymer Clay
Discussions & Questions board so they'll be sure and be seen by the polymer clayers!http://www.craftster.org/forum/index.php?board=239.0