There are lots of ways to make jump rings. Time is money, and it makes sense to buy base metal average size rings rather than make them. But sometimes you need a special size you can’t buy or you just need a few sterling or gold rings and don’t want to pay inflated prices for them..
The easy way to make them is with a dowel or mandrel, wrap a coil and cut it with a saw. You could use double flush cutters-but they are hard to find. Flush cutters leave a pinch of wire on one side which means you spend more time filing down the rings.
 If you want to make rings in bulk, its better to get a tool. I got to use one and I’d LOVE the jewelry tools fairy to bring me one someday. The Jump Ringer Tool from Connie Fox is GREAT tool but pretty pricey! A more affordable tool is the Koil Kutter.  Either would be a nice addition to a jewelry studio work bench.
I am completely self taught in making jewelry so I had to learn the wrong way before I could discover the right way to do things. For making jump rings, I always find that I am out of a jump ring in the metal that I need when I need one, so I usually make my own. I found as you indicated that making a jump ring using a mandrel or dowel and with one cut with a flush cutter did not work because one side was pointed and one side was flat. I now make my jump rings with two cuts for each jump ring. I begin by making a coil on my step jaw pliers and cut the first 1/8″ off the coil, leaving a flush cut on the coil. Then I reverse my cutter and cut the second time at the point on the coil where the wire completes the ring. This leaves a flush cut on both sides of the jump ring and a pointed end on the coil. Making the second jump ring requires two cuts with my flush cutter again.
Sincerely,
Gary Helwig
http://www.wigjig.com
Actually the simplest way to do your own jump rings is to cut them with a micro handsaw. That gives you no sharp ends or points