How do Buddhist monks find reincarnations of previous monks while filtering out the wrong persons? I'm skeptical about this actually working. I would appreciate if someone could provide a logical answer that will clear up my skepticism.
Sometimes it doesn't work very well: there are examples of reincarnations who behave badly.
And they must make it work: for example the current Dalai Lama wasn't keen to study, when he was young, his teacher had to make him study by threatening him with a whip.
At Home With the Dalai Lama

I also know that he wasn't a particularly good student when he was young. He had a mercurial temper and was impulsive. Monastic disciplines like meditation and scriptural study did not come naturally to him.
&quot;Around seven or eight,&quot; the Dalai Lama told me in an earlier meeting, a mischievous gleam in his eyes, &quot;I had no interest in study. Only play. But one thing: my mind since young, quite sharp, can learn easily. This brings laziness. So my tutor always keep one whip, a yellow whip, by his side. When I saw the yellow whip, the holy whip for holy student the Dalai Lama, I studied. Out of fear. Even at that age I know, if I study, no holy pain.&quot;
Despite his reluctance to study when he was a child, the Dalai Lama applied himself every morning. With perseverance and self-control, he learned to sit still for long periods. Gradually he was better able to control his errant impulses. Meditation and study came before play; delayed gratification became a matter of course.

See also Training.
Wikipedia has a description of how they found him, Searching for the reincarnation, which includes,

The High Lamas used several ways in which they can increase the chances of finding the reincarnation. High Lamas often visit Lhamo La-tso, a lake in central Tibet, and watch for a sign from the lake itself. This may be either a vision or some indication of the direction in which to search, and this was how Tenzin Gyatso was found.
It was here that in 1935, the Regent Reting Rinpoche received a clear vision of three Tibetan letters and of a monastery with a jade-green and gold roof, and a house with turquoise roof tiles, which led to the discovery of Tenzin Gyatso, the 14th Dalai Lama.[26][27][28]

There's a description of the theory and practice of it here on the Dalai Lama's web site: Reincarnation