Once flattened, the flattened overlap needs to be punched to accept a rivet. Usually it should not be necessary to re-anneal the rings prior to this step.
A primitive way to do this is to construct a simple hand punch, and punch through the ring into a block of metal with a hole in it (a punch block). To construct the punch, you will need a hard piece of steel (like a pin punch, available from Walmart), a grinder (like a Dremel tool with a diamond-coated grinding bit), and water (to keep the punch cool as you grind it, so as not to anneal it from the heat generated while grinding):
This is the shape that you want to achieve for your punch bit to make holes for wege rivets:
To use the punch, simple drive through the ring overlap into a small hole in a block of metal. The punch should just barely break through the opposite side of the ring, and you will be able to feel this happen:
A more sophisticated (but faster) way to do this is to construct punch tongs. I made a set out of tile nippers, screws, and drill rod. You will need a tap-and-die set (available from Walmart) and probably a drill press (though I have done it without one). NOTE: details on constructing these tools can be found in the Riveted Maille Index.
Once you have constructed your tongs, they are very easy to use to consistently punch holes in rings:
