Physiological and somatic theories of personal identity are, in general, quite firm on what happens at death; the person dies. End of story. It doesn't matter whether the individual theory claims that the brain or organism as a whole bears the label "person"; if there is physical death of that thing, then the person dies with it.
Psychological theories, however, can respond in a more complex way.
The first example that sprang to my mind was Douglas Hofstadter's view of the soul, which he explicates in I Am A Strange Loop, the "sequel" to his much-vaunted (and much misunderstood) earlier work Gödel, Escher, Bach: An Eternal Golden Braid. Hofstadter's soul, it must be noted, is not the standard dualistic fare;
"The central aim of this book is to try to pinpoint the nature of that "special kind of subtle pattern" that I have come to believe underlies, or gives rise to, what I have here been calling a "soul" or an "I". I could just as well have spoken of "having a light on inside", "possessing interiority", or that old standby, "being conscious"." (p. 23)So in Hofstadter's view saying that something is "ensouled" is simply a rather poetic variation of saying "this thing possesses consciousness" - and certainly not a claim that there is some form of non-physical substance attached to the thing in some manner. I admit to not having made the time to read GEB fully yet, but I would doubt that his views differ substantially between the two works (if, indeed, he discusses souls at all in the earlier book).
The relevance of I Am A Strange Loop to this post is that Hofstadter argues that our souls can live on after death; he begins with a discussion of how Chopin's music allows a small fragment of Chopin's mind to live on after his death, but the primary case study of the book involves Hofstadter's wife, Carol, who died suddenly of a brain tumour. Hofstadter argues that aspects of her soul lived on in his mind after her death; they shared a number of their desires and beliefs, and many years of memories.
This is a reasonable claim to make, yet it should be pointed out that by necessity Hofstadter and his wife did not share all of their mental states. Those mental states representing the body (Damasio's "proto-self", roughly) are an obvious example of such unshared mental states. The question that should be asked, then, is; how many shared mental states were there, and how significant is the loss of those that were not shared to the continuation of his wife's personal identity?
I'm not immediately sure how to answer those, nor am I in any frame of mind to adequately do so currently. My intuition is that, although it would technically be possible for there to be enough psychological connections between two individuals for the death of either not to matter in terms of psychological continuity, that such an event rarely (if ever) occurs. Although some of the deceased individual's psychological states will - almost inevitably - be continuous with the psychological states of others, there won't be enough of the deceased individual left, as it were, to claim that the individual pre-death is psychologically continuous with the "fragments" in others' minds post-death.
All that remains are aspects, small pieces, of the individual's self. That might change in future - advances in science could lead to methods of "saving" the majority of an individual's self post-biological death, such as the ever-popular "mind uploading" concept - but for now, I can't help but think that it is a sad fact of life.