Who owns the U.S. Census records? If I have a copy of the U.S. Census can I charge you to look at it? Who owns the Social Security Death Index? Again, if I have a copy, can I charge you to look at it? Do the individual states own their vital records? Do churches own their membership records? All of these questions have different answers depending on specific circumstances. Some of the answers involve copyright law and some involve practical business interests.
Starting with the U.S. Census, works produced by the U.S. Government as not available for copyright protection. However, the United States Government is not precluded from receiving and holding copyrights transferred to it by assignment, bequest (gift), or otherwise. See Title 17 of United States Code, Chapter 1, Section 105. For that reason, you can copy the U.S. Census records and reproduce them without violating any copyright held by the U.S. Government. Then how can someone sell access to the U.S. Census? The answer is simply that an index may be copyrighted. The business reason is also simple, people will pay for the convenience of looking up the records. But if someone chose to put the U.S. Census online without an index, it could be done without violating anyone's copyright.
Now what about the Social Security Death Index (SSDI)? It is already an index and comes from the U.S. Government. Could I charge you a fee to look at the SSDI? Yes, and if you are willing to pay me, I might make some money. But there are "free" copies of the SSDI on the Internet already.
The issue of the ownership of state and local records is more challenging. Generally, just as with U.S. documents and records, the states do not have copyright protection. But, generally, states charge for copies of the documents and records in their possession. Only a few states, for example, have put any of their birth and death records online, and their is almost uniformly a charge for their reproduction. But once you have paid for a copy of the record, a birth certificate for example, you could use a copy of the certificate in a publication without worrying about copyright.
European jurisdictions vary in their attitude towards the records created by the governments. In some countries, like Sweden, for example, the Government charges for access to its record copies. In Denmark, on the other hand the records are generally free and online. There seems to be no consistency in whether a given type of record will be free or subject to a fee payment for a copy.
What about old private records? The U.S. Copyright Office explains, "You can register copyright in [an old record] only if you own the rights to the work, for example, by will or by inheritance. Copyright is the right of the author of the work or the author's heirs or assignees, not of the one who only owns or possesses the physical work itself. See Circular 1, Copyright Basics, section “Who Can Claim Copyright.” A work that is out of copyright, either due to age or use, is said to be in the "public domain." It is best not to assume that a work is not copyrighted. Even old works may be subject to some one's claim of ownership.
For more information, see Copyright Basics from the U.S. Copyright Office.