Technically, the Neocatechumenal Way does not have public confessions. Members are asked to share their "crosses". Think of it is as an AA meeting. There is nothing wrong with this of course and it can be, as the AA has proved, the first step towards healing.
The problem is that most people's "crosses" are often their personal sins. There is the additional problem of a cross also being the sins of others, such as an adulterous spouse. So you can see how quickly sharing a "cross" can become a revealing of ones deepest personal sins as well as the telling of the sins of others.
Such a thing is extremely dangerous. It works in the AA because usually there is a skilled counselor in control of the session and everyone's "sin" (alcohol addiction) is already known, which is why they are there in the first place. But in the NCW, we are taking people who for the most part only signed up to learn more about their faith and suddenly find themselves learning about the deepest personal sins of others.
The pretext for this is probably James 5:16 where it says "confess your sins to one another". However, the preceding verses put this exhortation in the context of "summoning the presbyters". Thus we are not to confess our sins to just anyone. Logic should tell us that anyway. We are to confess to the presbyter.
And to emphasize this, the exhortation to confess in James is followed by: "the prayer of the righteous man availeth much." The idea, being of course, even in that primitive context, is that we should confess to one who is righteous, and by design, it is supposed to be the presbyter/priest.
As a side note, this public sharing of crosses is nothing new. The eminent psychologist Carl Rogers developed it in the 1960's as part of his system of self-actualization therapy. Curiously, Rogers ended up using an order of nuns in the diocese of Los Angeles for his first large group test. The results were disastrous.
Within one year, over 300 of the original group of 560 nuns were petitioning Rome to be relieved of their vows. And by the end of the second year, though the experiment had been designed for three years, Rogers himself, alarmed at the results, terminated the experiment.
In 1990, William Coulson, who was Roger's chief of staff and, at the time, in charge of the experiment, gave a talk to members of the L.A. diocese wherein he explained why the experiment was a disaster and publicly apologized for taking part in it. His talk was recorded and is available here. You can also learn more about the experiment in a piece I wrote here.