The silver core is really just a large rivet. But I know from experience that treating this process the same way as adding a rivet to metal is a mistake; the length of the tubing that runs through the bead makes it prone to buckling and the hammering puts a lot of stress on the bead and it can break or crack.
Someone (sorry, I can't give credit because I don't know who it was!) posted in a forum on Lampwork, etc about how to make your own tool to add the silver core to a lampwork bead. I followed his basic instructions, and it didn't cost me anything at all because I had the materials laying around in the garage already.
I started with a large nut and bolt--this one is two and a quarter inches in length and a quarter inch in width on the threaded end.
I took the nut off of the bolt and measured down the threading about a quarter of an inch. That would give me enough room to use with a larger bead--one I would use for a focal bead. Then I used a bastard file to take the threading off and fit the silver tubing I had.
That's it--the whole tool is a nut and bolt with the threading filed off.
Here it is in use:
First I measure the amount of silver tubing I will need to cut using the tube cutting jig:
I put the nut and bolt combination into my vise and added the tubing, then the bead.
The nut cradles the bottom of the bead to help keep it from breaking during the riveting.
It's still easy to chip the bead around the rivet, especially when the bead is off the nut/bolt and flat on a stainless steel block putting in those last few taps. If the hammer comes down a bit off center, it can take little chunks out of the bead close to the rivet.
The final product: