The glue will only be as strong as the paint or finish that is applied. I just had another customer ask me about this earlier today. I think it depends on what you are gluing. For instance, I would use a 2 part epoxy to glue a sawtooth hanger on the back of a plaque. (More than one if you think the plaque is a little heavy.) But if you are gluing say an 1/8" scrolled piece to a back board or another piece similar, where there won't be much stress on it, any good clear drying wood glue will probably do fine. I like Gorilla Wood Glue. I also like Tite-bond.
I think the most important thing is to be sure that the finish or paint is completely cured before applying glue. If you rush and glue it to fresh paint or finish, it could easily pull off of the piece (taking the paint or finish with it!) and come apart.
I am sure there will be some other great advice to follow, but that is my take on it.
Take care, Sheila