MTW has a discussion of this on p. 425. They give a final interpretation by Deser, with a reference to Deser, S., 1970, "Self-interaction and gauge invariance," Gen Rel and Grav 1, 9. They describe an interative process in which flat-spacetime gravity, which is not mathematically self-consistent, is repaired. After an infinite process of iteration, we arrive at a theory in which the original flat spacetime "is no longer observable." They claim that the resulting theory makes the same predictions as GR.

Mentz114 linked to a review article by Baryshev that is more up to date than MTW:

http://arxiv.org/abs/gr-qc/9912003 . Baryshev says that the two approaches are not in fact equivalent, so "field-theory gravity" (FTG) is not equivalent to GR as claimed by MTW.

Some quotes from the Baryshev paper:

"In fact, Deser showed no more then it is possible to find such an expression of EMT which at

the third iteration gives Einstein equations and in no way this leads to the conclusion about identity

of field and geometrical approaches, as was claimed in the book of Misner, Thorne, Weeler (1977).

Moreover, the essence of field approach suggests such a choice of gravitational EMT that satisfies zero

trace (massless graviton) and positive energy density of gravitational field and namely these properties

should be tested first. Deser’s EMT does not satisfy these conditions. It is easy to demonstrate that

positive energy requirement leads to radical difference of field approach from that of geometrical one

(see 5.3)."

"[...]black holes are prohibited [in FTG] by the energy conservation" (5.3)

"Frequently

one finds in literature that black holes have already been detected, because there are systems with

components more massive than the Oppenheimer-Volkoff limit, i.e. over the three solar masses. This

statement is not correct, since this limit exists only in GR, but in FTG there could exist relativistic

stars with larger masses."

What I'm unable to gauge at this point is whether Baryshev's interpretation is controversial, or whether everyone in the field now agrees that Deser's conclusion about the equivalence of the two theories was incorrect. A shortcut method for checking into this without digging deep into the math is to see whether Baryshev's review paper was published in a refereed journal (apparently not) and whether Baryshev publishes regularly in the usual journals where relativists publish (looks like the answer is no, after a quick dig through arxiv and Baryshev's web site). So I would be cautious about assuming that Baryshev is correct and/or noncontroversial.