The orthodox Trinitarian works that I've read all deny that there are three "I thinks" in God. Granted, the Trinity doctrine may allow room for the "God-man" Jesus Christ to possess a human consciousness and a divine consciousness simultaneously, but if we prescind from the Incarnation, then I think you'll find that Trinitarians generally do not believe there are three consciousnesses in the Godhead. This belief would normally constitute tritheism:
"The philosophical input from Descartes, Kant, and John Locke (1632-1704) led to the emergence of a (but not THE) typically modern notion of person as the subject of self-awareness and freedom--in brief, person as a conscious and autonomous self ('I think and am free, therefore I am a person'). This notion, when applied to the doctrine of the Trinity, readily produces what looks suspiciously like tritheism: three autonomous subjects living and working together in a quasi-social unity" (Gerald O'Collins, The Tripersonal God, 155-156).
"In contemporary parlance, person is spontaneously identified as centre of consciousness and freedom. However, if we bring these pre-reflective categories to theology, we are immediately confronted with a problem. For if we say that God is one being in three persons, and if we understand by person centres of consciousness and freedom, then God becomes three centres of consciousness and there are three I think's in God. But such an understanding is the same as tritheism" (John J. O'Donnell, The Mystery of the Triune God, 103).
So O'Donnell attempts to effect a synthesis between the differing trinitarian notions of person. He observes that trinitarians have either viewed the tres personae as distinct subsistent relations (Karl Rahner), or interpreted the "persons" as separate centers of consciousness. As an alternative to the previous accounts, he suggests viewing the triune God as "one divine consciousness . . . shared by three persons" (110). Analogically, this means that "whereas in human experience, person and community are not identical, in God they are" (110).
While O'Donnell's analogical language does make some distinctions not hitherto made by trinitarians in the past, its shortcomings are that it seems to employ meta-experiential and meta-rational methods to accomplish its task. If, as O'Donnell intimates, community and individuality are the same thing in God--is there truly an analogy of being (analogia entis) between God and man?
Addendum: of course, there is debate regarding whether there is one or there are three consciousnesses in the Godhead, but I often find that the one consciousness view wins out, for the most part.