Depends on the washer in question, the paint, how it's sealed, and the detergent. There's really a number of different factors that can affect that.
If I were you, I'd try running a "sample" of the studs (something of the same material or similar material painted in the same way) through the wash with the same conditions and see what happens. If you know the paint's not temperature-sensitive (it's not the same, but you could always try putting a little boiling water on a stud to see how it reacts?), then I'd wash your stuff without detergent (or something very mild in its place), or even omit washing and just get it damp, put it into a delicates bag to protect the studs from whacking against metal, and run it through the dryer on high several times instead.
(Note that this is all speculative. I've always made those sorts of things removable so that I could still wash the garments.)