You are opening a rather complex conversation Ganderite. First consider what causes the double gun to loosen up and which of these problems is in play on any particular side by side.
Most guns today are held closed by the Purdey style double underbolt on the lumps ( protrusions) on the bottom of the barrels. Notches in these under lugs are engaged by a sliding under bolt in the action table, holding the barrels down to the table. This prevents the gun from popping open upon firing and it can be in conjunction with various forms of a top bolt such as the popular Greener top bolt, Westley Richards top bolt, doll's head and some other methods that work on a top rib extension. In testing it was found that flexing of the action upon firing causes a momentary gap, no matter thhe closure method and also that a properly fitted top bolt is sufficient by itself to leep the gun from popping open. Thousands of Early Westley Richards shotguns and rifles with only the top bolt still lock securely securely after 140, 150 and more years of service.
Another strain on side by sides comes as a llateral left or right twisting caused by the firing pressure being on one side or the other of the bottom locking bolts and top bolt if present. This lateral thrust is addressed by close fitting of the rear faces of the barrel breeches to the standing (vertical) face of the action opening. It is usually aided by the close fitting of the bottom lugs into their recesses in the table ( horizontal bottom portion of the action body).
The third force acting to loosen the action is direct backward thrust backwards of the barrels against the standing breech.
These three forces all act together to loosen the gun every time it's fired and once any looseness develops it will worsen with use. Heavy loads in these conditions will loosen the gun quicker and the looser it gets the quicker it progresses, like a roll of toilet paper unwinding. To prevent this from happening for as long as possible......
Assuming a sound design, the correct materials, each properly hardened to the ideal for it's particular purpose and very closely fitted upon closing the gun is the basis for for a solid long lasting gun. Of course, excessively heavy loads and ammunition too long for the chamber strain the gun and cause it to loosen sooner. Most early Belgian hardware store guns fail on most of these points. How to tighten and put a loose gun back on face?
First you need to determine what is causing the looseness, often it's a combination of different forms of looseness but there is no one fix for all forms of looseness although they often overlap.
Up and down looseness, not locking firmly down is quite rare by itself on English guns. The simple and dirty fix as shown on the Potterfield video consists of hammering the back face of the rear locking lug to peen material upwards, minutely closing the locking gap on the rear lug and then filing it in to fit. The added bonus to this method is that it also somewhat closes the sides of the table recess which helps with side play. It's quick, it's easy (comparatively) and it's cheap. BUT it doesn't last because it doesn't address the gap looseness at the other (front) locking lug and it does nothing for fore and aft looseness. It is not reccomended as a sole method in most cases because......
Most often there is looseness side to side and especially between the rear faces of the barrels and the standing breech ( the part with the firing pins in ). Then the barrels need to be set back and tightly and carefully fitted to the standing breech. This is accomplished by building up the front face of the hook ( the semi circular indentation that fits onto the hinge pin forming the hinge) and then filing it in to a perfect fit to bring the barrels back to contact with the breech face. For generations this was done by filing or machining a notch into the front surface of the hook, pressing in an oversized piece and then filing it in to get that perfect breech fit. With modern technology this hook is more commonly built up by TIG welding ( can be done without melting the solder fastening the hook to the barrels) and then filing in. The latest advancement here uses spray welding instead of TIG welding. Now all of this is time consuming and costs hundreds of dollars but the gun will lock up like new and it will last. One important note, if a gun has a third bite ( top bite) like a Greener cross bolt or Westley Richards top fastener it needs to also be perfectly fitted as well to spread the load, this adds considerable complexity to the job ($$).
So there you have it, it takes years of experience to determine how to rejoint each particular gun and to do this critical job properly.
J.