A professional will use a special fixture to lock the barrels and ribs together so when they heat the barrels to soft solder it, the barrels and ribs can't separate.
This does not mean they'll just wire everything together and hope they stay together.
Barrels have not only two barrels, they also have several ribs all soft soldered together.
These barrels and ribs are usually under at least some stress and tend to "pop" apart when you try to re-solder them.
Since better guns have the barrels regulated or adjusted to shoot to the same point during manufacture, once they come apart they will no longer shoot properly.
That's why you need a real professional shotgun expert to re-solder barrels, not you local gunsmith.
Next, since there's rust present and solder will not adhere well to rust, a professional will have to know how to deal with the rust issue.
What solder used depends on the gun and it's age.
Older gun makers used almost pure lead as solder, later makers used a more modern soft solder, each using a different type according to their experience on what worked.
A few modern guns may use high temp brazing instead of soft solder.
Bottom line: This is absolutely NOT something you want to try on your own unless you're an expert shotgun barrel technician.
Try it on your own and either the barrels and ribs will pop apart, or worst, they may not pop apart......until you shoot it, and possibly not until you've shot it a bit and think you're good to go.
In either case, the charges to totally disassemble and completely re-assemble the barrels correctly so they shoot together will give you a bad case of money shock for the charges.
Best advice: Be smart and send them in to a real expert like Briley or Simmons.
This is one of those gunsmithing jobs that really does require a genuine expert specialist.