I've converted to all bolts and nuts but I can tell a 9 year old how to do it for rivets.
My 6" vise is on a stand at elbow height. I have a couple of outboard stands like you would make from old floor lamps to support the end of the knife bar as it moves along the vise.
Take the cutter bar with tips down and put it in the vise so that the knife bar rides on top of the vise jaws and a portion of the section is down within the jaws. You want no angle on the bar, let it rest squarely on the vise jaws. That means horizontal. Tighten the vise just a tiny bit. This will keep the knives from flying away when they separate. The bar should easily rock in the vise. That's why you may need a helper or a support stand.
From the left side view, you have a jaw on the right, the bottom edge of the cutter bar riding on the top of the jaw, a sickle section running down through the vise, and the back (opposite) jaw. The vise is tightened ever so slightly.
The sickle sections on cutter bars usually line up so that about 1/8 to 3/8 inch of section is overlapped. This amount is affected by wear and how good your hold down clips are. This means that if you can give the top edge of the section a large wack with a heavy square face hammer, it will easily shear a rivet because its hard steel. The bar is supported by the vise jaw, and not receiving any hammer force (its 1/8 to 3/8" below the section) so its not going anywhere.
As you get used to the process, you will find that a good square hit on the section can shear both rivets. The section will drop through the vise and the bar will still be supported by the vise jaw. Having a vise as wide as several sections will keep the cutter bar in place on top of the jaws as you knock one down at a time.
When you get the hang of it, loosen the vise so that it can slide along the vise jaws. That will let you pound off each and every section with smooth and continuous strokes. You are not swatting mosquitoes here, pretend you are driving 20 penny nails into oak and want it done with 5 blows (just use one though).
This works for just one sickle section or all of them. Make sure you wear safety glasses because rivet parts will be flying. That goes for helpers, too.
You might need to drive out short pieces of rivet from the bar with a center punch because the shearing action will occur only at the bar to sickle interface. The bottom end of the rivet will most likely remain in the bar. That depends on who put them in and cold or hot.
Fun, but not easily done in the field if you break a knife. That's why I switched to bolts and nuts. You can imagine doing this with 2 pieces of angle iron and some C clamps, but the heavy mass of the vise ensures that the soft rivet takes all the strain and shears through.
