Obviously, the historical sources of the text(s) have no copyright and anyone can publish a Bible. Usually, however a company is set up to market a particular translation. In the case of The New International Version (NIV) for example, which a relatively new translation of the Bible made by more than a 100 scholars working directly from the best available Hebrew, Aramaic, and Greek texts, the corporate body engaged in the actual translation and publication work is The International Bible Society (IBT), which is the translation sponsor of the New International Version� (NIV). The Committee on Bible Translation (CBT) is an international, non-denominational working group of scholars, historians and linguists engaged by the IBS. The CBT produced the translation after nearly 20 years planning and work and then had the publishing rights to that particular book. It was well received by most evangelicals and the profits received, after paying enormous expenses have largely gone into further research as well as publishing a Children's bible in NIV. The NIV was published in the U.S.
Although the NIV is well received by many denominations, it isn't the intellectual property of any one church body, but rather that of the IBT. Anyone may use it's content, but, of course, as with other copyrighted publications, it may not be printed verbatim and sold commercially...