Sometimes copper pots come to me with loose rivets and they really need to be tightened during the cleaning and prepping stage before I put the tin on if I’m going to do the job right.
However, you don’t want to “fix” the loose rivets by whacking on them with a hammer. That can do a few things: it can mushroom more rivet material between the handle and the pan, worsening the problem. It can make the rivet crack (especially older, brittle copper rivets), which would then require the entire rivet be drilled out and a new one sourced (!!) and then re-riveted into the hole. Neither situation is desirable.
So in order to make your rivets tight, you want to first get a piece of metal or wood that is hollow enough to enclose the rivet on the handle. This way, there is no direct pressure directly on the rivet itself but on the surface of the handle (and, subsequently, on the metal pot’s sides below the handle) surrounding the rivet. Then, you’ll want to find a block of metal or hard plastic that fits inside the pot and is relatively close to the radius of the interior of the copper pot’s diameter. If the interior is nice and newly tinned or has a stainless or silver interior lining, and you’re doing this yourself, put a cloth over the metal or plastic before you place the pot’s interior on it to protect the interior lining. But if you’re restoring it, and the pot has been cleaned and stripped and doesn’t have the new tin yet, you don’t need the protective cloth.
Hold the pot steady, with the rivet area you’ll be hammering directly in the center of the interior fixture holding the rivet in place from the inside. Place your piece of wood or metal over the rivet on the handle, and hammer hard a few times.
Check and repeat as necessary to both rivets.
The trick is to make sure there is a good back stop holding the interior rivet, and a piece to hold over the rivet itself on the handle before hammering the material tighter.