I have found that the older 24's with the barrels soldered together are very accurate. I expect 1/2" to 3/4" at 50 yd. groups from the .22 barrels. The newer ones where the barrels are separated and clamped at the front are generally not as acccurate, but should shoot under 1" at 50 yd. The centerfire rifle models in my experience are not as accurate, but I have come to expect 2" groups at 100 yd. I have a .222/20 ga. 24V that will group 1-1/2" at 100 yd.
I can't speak about shotgun slug accuracy except with a newer .30-30/12 ga. that grouped the slugs into 3" at 50 yd.
The biggest problem with 24"s is, as Pastorp stated, both barrels not shooting to the same point of aim. Some are pretty good, whild others are pretty bad, with the worse problem being a large difference in elevation and windage. This can, in some cases, improved with reloading and/or trying different factory loads. I tend to buy them, try them, and replace the ones that aren't even close.
My current crop consists of: .22/.410, .22/20 ga., .22 mag./20 ga., .222/20 ga., .30-3-/20 ga. The .22/20 was purchased here at GBO and hasn't been tried out yet.
The problem with shooting the shotgun barrel with a scope attached can be taken care of by removing the elevator from the rear sight and placing the forward scope ring against the sight. This keeps the scope from moving forward when the shotgun is fired. I don't know how this will hold up in the long run since I haven't fired a lot of shotgun shells thru any of the guns. The centerfire guns do not have this problem since they accept Weaver mounts.