The thing that you have to remember is that a private revelation is, first and foremost, a private revelation. Which means that it is specifically a message for the one(s) who receive it directly.
An approved private revelation is one in which the Church has found the message may enhance the spiritual lives of other people (therefore, one may have a devotion to–say, Our Lady of Lourdes), but since we are neither Bernadette nor other individuals of Lourdes in the 19th century, we are not obliged as St. Bernadette was. On the other hand, if that particular apparition does nothing for you spiritually, you are not obliged to make any recognition of it (other than not denying it)–at least, unless you happen to attend a parish named Our Lady of Lourdes, in which case you’re going to probably encounter more than average devotion to that particular apparition. (The same goes for any other approved apparition, just change the names to Fatima/Jacinta, Fernando, and Lucia; or Guadelupe/Juan Diego; and so on.) For these people, belief in the apparition was not an option; for everyone else, it is. Like the Rosary–no one is OBLIGED to say the Rosary (at least, prior to joining a Confraternity, but such action is voluntary itself, and failure to say the Rosary even then isn’t a sin), but millions have found it enhances their spiritual life.
An on-going thing is not going to be approved. No one outside the one receiving that vision should maintain any sort of devotion to it. A false apparition should not have a following at all.