Are brake drums (in this case the rears) balanced from the factory? If so can they be balanced again if you think one's out of balance?
I know, stupid question I guess. Just wondering 'cause one of mine has (what looks like) a big weight spot welded on it. The other doesn't

There are no stupid Q's. I always assumed that was a spot welded balance weight as I cant think of any legitimate purpose for it to be there, doesn't look it could be a locating tab for machining of some sort like you see on blocks etc plus the other drum has nothing on it and cast parts are nowhere near concentric or any part really. Not sure if it could be balanced like on a brake lathe with a strobe light or.... The tire guys may chime in but I'd think spin balancing the whole tire/wheel assy on the car with one of those gismos that you put under the tire & it spins the wheel would balance the whole shebang. Myself I'd rather have the drum and the tire/rim assy all neutral seperately then together to check so I could switch tires/rims around if I so desired & have it dead on

I've noticed some have weights and some do not too, I think its because the balance spec (+/- x amount) is pretty forgiving due to the relatively small diameter at play compared to the wheel/tire. A small amount of imbalance would likely never be noticed, therefore many drums didn't get/need the balance weight.

If I had a dollar for every drum the weights came off of in 40 years of wrenching Id be as rich as
Trump.Cant remember how many dropped off trying to get stuck drums off the hubs.Never any problems.There are more important things to worry about than that.Rocky

I guess the premium drums are balanced and the "value brand" are not. I've never heard of balancing them after purchase.

If you can't clear up the shakes, however, on car tire balance would take care of it. The balance job would cover the total wheel/tire/drum as assembled.