Rings of Brighthearth can indeed be used to copy your planeswalker activations, but doing so won't put any more loyalty counters on those planeswalkers.
When you copy an ability, you place a copy of that ability directly onto the stack, without paying any of the costs that would normally be associated with activating it, in the same way that if you Twincast your opponent's spell, you don't also have to pay the cost of casting that spell.
Putting the counters onto a planeswalker to use their positive loyalty abilities (
, 
, and so on) is one of the costs of activating the ability in the first place--it's not part of the ability's effect. So copying the ability bypasses that cost--you get the effect a second time, but you don't also get twice the counters.