The Church has never proscribed private possession, reading, or translation of the Bible (and indeed the Douay-Rheims Bible, long the standard Catholic English translation, was published before the King James Version).
The Church was very concerned in the mid-16th century that Protestants had made their own decisions about what books to include in the canon and how they might be interpreted. In the fifth session of the Council of Trent, the Council made formal declarations on these subjects.
No one, relying on his own skill, shall... presume to interpret the said sacred Scripture contrary to that sense which holy mother Church,--whose it is to judge of the true sense and interpretation of the holy Scriptures,--hath held and doth hold; or even contrary to the unanimous consent of the Fathers; even though such interpretations were never [intended] to be at any time published. Contraveners shall be made known by their Ordinaries, and be punished with the penalties by law established.
And wishing, as is just, to impose a restraint, in this matter, also on printers... [this Synod] ordains and decrees, that, henceforth, the sacred Scripture, and especially the said old and vulgate edition, be printed in the most correct manner possible; and that it shall not be lawful for any one to print, or cause to be printed, any books whatever, on sacred matters, without the name of the author; nor to sell them in future, or even to keep them, unless they shall have been first examined, and approved of, by the Ordinary.... As to those who lend, or circulate them in manuscript, without their having been first examined, and approved of, they shall be subjected to the same penalties as printers: and they who shall have them in their possession or shall read them, shall, unless they discover the authors, be themselves regarded as the authors. And the said approbation of books of this kind shall be given in writing; and for this end it shall appear authentically at the beginning of the book, whether the book be written, or printed; and all this, that is, both the approbation and the examination, shall be done gratis, that so what ought to be approved, may be approved, and what ought to be condemned, may be condemned.
The Bible (like other books specifically thought of as Catholic) may be printed, circulated, read, and translated. But before they are, they must be examined, usually by the local bishop's office, and signs of approval must be printed in the book. Today these are known as the nihil obstat ("nothing stands in the way") and the imprimatur ("it may be printed").