I don't know how other people do it, but I have a pair of pliers in my right (dominant) hand, a pair of pliers in my left, the chain I'm working on in my left hand as well, draped over my index finger. I pick up the rings with my right hand and use the left pliers to grasp while I get my right hand pliers to open, link the ring into where I want it, then grasp the ring again with the left pliers to close. It sounds awful and you wonder where the extra left and right hands come from, but it works well and I never have to put anything down, unless I drop a ring. Basically, I hold the weave the entire time as well as my pliers and use my fingers only to pick up rings and move the links on the weave. If I'm doing a really tight weave and I need to use an awl, that's about the only time my right hand pliers are put on the table.
I'll get my partner to take a series of photos next time I'm doing something so you can actually see what I mean.....I don't know that it makes much sense the way I explained it.
If you like opening the rings with your fingers, why not get a ring-ring opener? It's a ring that slips over your finger with slits in it that you use instead of pliers. I wear mine as an everyday ring, but I have VERY small fingers (J-K size in rings). If you have larger fingers, it will sit above your knuckle.
I love fresians and andalusians, we have a couple of students at school who breed the andalusians and I always love seeing the photos when they foal. Where do you sell online and I'll direct them to your site, you may pick up some business?