To take the example you just gave of a kidney, this kind of thing would be ideal if renal failure were caused by, say, overuse of aspirin / ibuprofen / any one of a collection of other drugs, or even if it were the result of infection by, for the sake of argument, hantavirus if the infection could be completely contained or, ideally, eradicated ahead of time. They grow an undamaged kidney using the patient's cells, perform a transplant as usual and hey presto, as best I can tell.
If it's the result of a genetic defect directly causing the problem, it'd be different -- they could grow a kidney from the patient's cells, put it back in, and while it might work for a time depending on the nature of the defect the fact remains that the cause of the renal failure / disease / whatever is still there. If the regrown kidney were to undergo gene therapy that remedied the defect, however, it's possible that it'd then be 'safe' to introduce to the patient's body.
That said, I'm not a biologist; everything I said could have been complete nonsense (though it seems to synch fairly well with what I'm reading about it).