Defending the idea of a Trinitarian God is such a difficult task, that I am almost sure I would never try. Nonetheless, I can sympathise with theologians and philosophers who have tried to make sense of this conundrum.
The main problem involved is how to avoid the Scylla of Tritheism (i.e., believing in three distinct gods) and the Charybdis of self contradiction (when one claims that the Father is a different person than the Son, but that the Father is the same God as the Son).
One way to try to solve the problem is to deny that "being the same God as" implies identity. Authors like Peter Geach have tried to claim that there is no absolute identity and only "relative identity". This is an interesting idea, but it is hard to agree with it whenever one comes to our worldly experience (are we really ready to say that the fact that "The sunglasses I found in my bag are the same ones I have been looking for in my room" does not imply absolute identity between the two?). If, as suggested by Peter van Inwagen, relative identity only applies to particular cases, e.g., to God, then it is easy to rebut that the theory is only an ad hoc postulation.
Another solution would be social trinitarism, i.e., the idea that unity among the three persons of the Trinity exists only in a social way. Social trinitarism has been explained in different ways, e.g., as entailing that the unity of three persons of the Trinity is tantamount to their mutual love. However, it is hard to imagine a social unity which does not, at the same time, affirm an ontological distinction, so that the risk of tritheism is strong.
What else is left? Trenton Merricks (disregarding Moti's suggestion) uses a thought experiment by Derek Parfit to claim that brain bisection does not entail the creation of two distinct persons. Empirically speaking, brain bisection does now produce individuals whose left side ignores what happens in the right one and vice versa, so that one might be inclined to see them as two distinct persons, but, claims Parfit, this is not necessarily the case (in Kant's terminology, the fact that two persons seem to be there is not an apodictic judgement). For instance, one might have a device to block communication from one's hemispheres at wish and it would be difficult to claim that one person is created or destroyed each time. Instead, brain bisection "divides the consciousness of a single person", creating "two spheres of consciousness" (Merricks p. 310).
Similarly, Merricks continues, God, who is a disembodied soul, has distinct spheres of consciousness, though no brain. He (or She) is one substance, but three spheres of consciousness. The thesis has some explanatory advantages, since through it one can make sense of New-Testament passages where the Son speaks with the Father and seems to ignore what the Father does already know. Similarly, one might think that the Father has no complete experience of what it is to have a body, since He never had one. But what about the Holy Spirit? What would be His/Her/Its distinct sphere of consciousness?
More in general, is there any other way to lose the conundrum without going back to some (odd) metaphysical assumption? What do philosophers of religion on the Cocoon think?
(Peter Geach, Logic Matters, 1972
Trenton Merricks, Split Brains and the Godhead, 2006
Derek Parfit, Reasons and Persons, 1984
Peter van Inwagen, And yet there are not three Gods but one, 1995)