There are three possibilities for a modern mail-maker to make a hole for riveting into the overlapping area of a ring. The most obvious modern method is to drill a hole. I advise you to use high quality drill bits, which are sold for industry and workshops, not the usual ones from hardware stores. Beside this there are the traditional methods of piercing with an awl or punching with a punch. Especially for wedge rivets piercing is a good choice. I use an awl like the one in the picture:
I make the hole by hitting the awl twice with a hammer. A piece of hard wood lies underneath. I have to change it sometimes. You could also use metal to underlay, with an hole for the protruding point of the awl, but this can damage the awl very likely. The wooden underlayment even protects the awl from rings, that have not been annealed enough or from rings with a thick overlapping. Those rings will be pushed far into the wood.
You can attach underlayment and awl to a pair on thongs, as shown on mailleartisans.org. This is just worth the effort, if your point doesn’t break too often. I doubt that this was done back in ancient or medieval times. However the method of piercing can be as fast and exact as drilling and the resulting rings are very firm, because no metal was removed.
For round rivets I prefer punching the holes. This means to use a punch wit a flat side facing the ring. It removes pieces in shape of a circle from the overlapping area. I tried to forge my own punches by medieval methods two times, but it is difficult to adjust the hardness appropriately. It shouldn’t be too soft but also not too hard that it’s brittle. So I use punch from modern steel until I figured this out: