To clarify I am not asking How do Horcruxes work? To me the a horcrux seems to be the same as a phylactery which a lich would use (which I first read about in The Chronicles of Prydain) it is simply a way to keep a soul tethered to this realm but with a stronger link than say a ghost. (There is a chance that eventually Voldemort would have made himself into a being that could not be killed without destroying his horcruxes but that is merely conjecture.) Lilly Potter’s unknown mother’s love protection spell seems to have killed (by reversal or backlash) and split his soul into an uncontained soul shard and into Harry which means it was split into two non-official Horcruxes. This allowed Voldemort as a soul creature thing to regain strength by attaching to Quirrel and drinking Unicorn blood and so forth.
There is evidence that the horcruxes were designed to do more than just contain parts of his soul though. With the Riddle Diary it seemed to be recreating Voldemort as he was at seventeen by taking Ginny's life force. This means that his first horcrux was able to rebuild himself. I find it hard to believe that he wouldn’t make this a feature in all of his horcruxes. (It is my personal opinion that he intended to make a completion spell after creating his seventh official Horcrux and that would have made him close to immortal, something like a lich but without the rotting away.)
Voldemort meant to make 7 Horcruxes. Each one is a splitting of his soul that can tether him to the living world in case his current body is destroyed. Each one also seems to have a some aspect of personality either originally there or put into place by Voldemort. Now Voldemort is too smart to not have a way for each horcrux he made to have a way to recreate him or aid in his regeneration. Assuming that Voldemort wouldn't just make basically what amounts to save points without a way to regenerate. How exactly was he supposed to regenerate and was it different for each Horcrux?
I don’t believe there is a cannon answer so anything directly from the creator or statements made with facts to back them up will be considered. Pure opinion will not.
Also while researching question and answers for this question I've come across conflicting answers for what a horcrux does. So please if you feel there is a duplicate of any kind share a link and specify.