I have turned a bunch of stuff off center that made my lathe shake, even though it's bolted to 20,000 lbs of concrete, and did something like what JRJohnson did. I have a bunch of lead diving weights that I melted down and use to balance the load, however I first try to balance by bolting to my waste block. I would caution against using the tailstock if you can avoid it only to reduce the dynamic load on the headstock bearings.
Even if it's statically balanced, there will be a large load on the two headstock bearings in opposite directions if the headstock and tailstock have opposite but perfectly balanced loads. It might not amount to much turning at low speeds, but 2x speed is 4x the load, and just think: You're increasing the load on the bearings and the housings and bolts that hold them in place when balancing on opposite ends of the headstock, rather than eliminating the dynamic load by static balancing at the same end of the spindle.
I did a quick calculation on a theoretical lathe with 12" between bearings and 20" between head and tailstock (= 4" bearing to weights): the dynamic cyclic bearing loads with 60 lbs unbalanced (equivalent to about 18 lbs@8" from center at 240 RPM and increases proportional to the SQUARE of the spindle speed) are 20 and 80 lbs, and this increases to 100 lbs on each bearing if you balance at the tailstock. Having bearings farther apart from each other makes it a little nicer: unbalanced bearing loads if the bearings are 24" apart (same 4" between bearings and weights) is 10 and 70 lbs, static balancing at the tailstock would make it more like 80 lbs steady cyclic load on both bearings.
So it doesn't double the bearing load, but it does increase it significantly. Anything that can make the whole lathe move on the ground, I don't want that much load constantly going through my bearings and the bearing races & housings. It's going to fatigue and break them eventually.
If you balance on the same end of the spindle by putting weights on the wood, the unbalanced loads on the bearings will disappear (excepting the weight of the wood&metal due to gravity, which will not change with spindle speed).