The Ascian Prime and Estinien-Nidhogg are two different cases.

Dialogue before and after the Ascian Prime fight suggests they're violating the laws of the world to go into that form. Lahabrea states the power of the Echo (which all Ascians possess) can be used to break down the barriers of existence, while the Ascian Prime's dying scream is "What walls are these?! Damn you, Hydaelyn!" Both Lahabrea and Igeyorhm are also willing participants, so what we did was beat the Ascian Prime down until it couldn't maintain its causal violation and separated back into the two Ascians. Our Echo didn't factor into it at all, probably, maybe, I think.

Estinien-Nidhogg, on the other hand, seems to be a textbook case of possession. Nidhogg's lingering will is possessing Estinien's body and overriding his will (how it reconstituted his form I don't know, but hey, magic). It won't be as easy (or simple) as just beating him up until Nidhogg leaves Estinien's body; in fact, killing him may be part of Nidhogg's revenge in an effort to plunge us into further despair. I'm not precluding saving Estinien; in fact, I'm quite certain it's something many players want (and the WoL borderline needs at this point). Just sayin' it could end tragically just as easily.