Years ago when I was young and money was tight, I was springing for tires on my truck. The man behind the counter realized my situation, and told me that I could save some money by not balancing the rear tires... as solid axles do not need it... The downside would be that they could not be rotated to the front.
I took his advice, and it worked perfectly.
All new trailers come from the factory with unbalanced tires. I have never read here of any issues because of this fact. I have also never read of anyone taking their brand new trailer into a shop to have the tires balanced.... They just run them until it is new tire time, and then worry about it.
Having said that, I have since run them both ways with no difference in performance. When buying new tires, I cut my best deal and then at the last minute, ask for free balancing... They almost always agree. Once they did not. I drove away, got 2 miles up the road and my phone rang.... The tire shop, was calling wanting me to come back, as the free balancing was no longer a problem.
Balancing is a profit opportunity for them. They are paying the employees the same whether they do it or not.
As for how to have them balanced.... As has already been stated, most trailer rims are lug centric, not hub centric like most autos.
IOW, the center pilot hole is just stamped out roughly centered. So using this hole to mount them on a balancing machine will not balance them.. In fact it very eaisily can make them worse!
As far as the brake drums being unbalanced, I do not doubt this a bit, though I havn't verified it.... It does make sense though. The manufacturers will not pay for anything that is not needed. solid axles do not "need" balancing... If that works for the tires, then it will for the drums as well.
How many people balance their drums?