It is the speaker that is doing both the される and the たまらない in this sentence. (i.e. he is the one being passively subjected to the other person doing 自己完結).
In general I don't think you would go too far wrong to remember たまらない literally as "cannot bear" rather than something abstracted like "extremely". Most of its uses follow naturally from that literal definition.
三省堂 entry for たまらない 【▽堪らない】
So in this case, the speaker is saying that he really doesn't appreciate being 自己完結'd by the other person, and is hoping (asking) that they stop.
Also, be sure you understand what is meant by 自己完結. If you aren't completely clear about the sense of an unfamiliar Japanese word and it doesn't fit super-snugly into the context, consult a Japanese monolingual dictionary, especially for phrases like this when EDICT is more confusing than helpful with its short definitions. Concerning 自己完結:
(Regarding some matter, the state of being satisfied or reaching a resolution only within oneself. Sometimes used with a negative nuance of "though no resolution has yet been reached from the perspective of the people around him, he himself selfishly considers it settled".)