The Stalker can talk, since its being piloted by a Triarch.
Ghost/Doomsday Ark appear to be piloted by a Lychguard or Triarch, so I would assume they can talk too.
Scythes, Barges and Tomb blades are piloted by Warriors, so they can only moan or scream when they die.
It says in the Codex that Destroyers have to come from the upper ranks in order to have enough free will to become obsessed with death.
Actually, almost all necrons are suffering from Engramic Decay of some sort, leading to personality quirks ranging from those found in a nursing home to Arkem Asylem.
As for names, I'd say the elites(lychguard, triarchs, deathmarks) and up (lords, overlords, crypteks)retain enough of themselves to take names. Some may remember their fleshy names, other may just take one they like.
And I'm pretty sure Obyron's a full fledged Overlord.