I work with only my hands and pliers/ cutter. No welding or soldering is used. Most of the copper wire I use is stranded electrical wire that I strip from the plastic sheath and unwind to access the individual wires. When I'm sculpting the wires it feels like I'm doing a combination of drawing and sewing the wire together. It's taken many years of practice to perfect the wire connections so that they are secure and sturdy.
My first technique of wrapping shaped coils around a silhouette has evolved through trial and error to more elaborate and detailed fleshing out, but the spirit of the process has stayed the same. I imagine the object I'm creating is in front of me and I'm forming the wires over its structure. The final forms are quite sturdy because I frequently wrap and tighten the wires wherever they cross to create a mesh effect.
Here is a nifty scrap book page made by a lovely woman who caught me working on a couple of pet portraits at a Starbucks in Florida. She snapped a few photos, worked her magic, and Viola!: