Yes, there is a way of knowing; perhaps two ways. One would be from personal experience; another would be learning from those with personal experience.
From the Foreword to the Second Edition of the Big Book:
Figures given in this foreword describe the Fellowship as it was in 1955.
...
While the internal difficulties of our adolescent period were being ironed out, public acceptance of A.A. grew by leaps and bounds. For this there were two principal reasons: the large numbers of recoveries, and reunited homes. These made their impressions everywhere. Of alcoholics who came to A.A. and really tried, 50% got sober at once and remained that way; 25% sobered up after some relapses, and among the remainder, those who stayed on with A.A. showed improvement. Other thousands came to a few A.A. meetings and at first decided they didn’t want the program. But great numbers of these—about two out of three—began to return as time passed.
...
Those numbers above demonstrate a 75% rate of recoveries within AA prior to 1955. I've seen that same rate occur within AA since 1984, wherever people work the program as suggested.
If some other program could realistically claim a 10%, or a 5%, or even a 2%, rate of recovery, it would be making headlines.
Religions, if they could be honest, would be showing far below a 1% rate of recovery.
Those aren't real statistics. There aren't any because of the anonymity.