Prions are just proteins that can fold in a way that self-propagates by causing other instances of the same (or very similar) protein to also misfold the same way. This is how they cause infectious disease. The downstream impacts vary because it depends on what the misfolded protein actually does in addition to helping other instances misfold. Most often it just accumulates, avoiding destruction by the cell’s usual mechanisms of destroying misfolded proteins (most prions that can’t avoid this would be destroyed before we could even notice their existence). But even just accumulating a bunch of a prion can be devastating, as their mass can be disruptive and the actual functional amount of the normally folded protein becomes effectively nil.

But like all proteins, they are indeed still produced according to their genes. Proteins are generated through the transcription and translation processes.