Nobody. That is the very definition of public domain works. That having been said, there is a axiom in copyright with respect to such works -- "you can copy the original, but you can't copy the copy." In other words, if you obtain a copy of a public domain work, you may copy, scan, distribute, etc. it (like the works at Gutenberg.org), but you may not be able to go to your local book store and buy a copy of "Moby Dick" or "Tom Sawyer" and scan or photocopy it, notwithstanding the fact that the underlying stories are in the public domain because, unbeknownst to you, aspects of the bookstore version may themselves be copyrighted (i.e. the cover, accompanying notes and commentary, illustrations).
If you are considering doing anything like this, you should obtain the guidance of a copyright lawyer for the specific copy of the work you are interested in.
That depends. There is ongoing litigation about the use of high-resolution scans that were paid for by public libraries and museums -- many institutions will allow use of the low-resolution copies but require payment (i.e., a de facto licensing fee) for the high-resolution images they make available online.
If you want to be safe, go to the museum yourself, and take a picture of the public domain work with your own camera. You own the copyright in your photo, and can do whatever you like with it.
The scans in various circumstances can be considered derivative works. So as my colleague state, even though you might be free to copy the original, you should not assume that you can copy the "scan" of the original. The scan should be deemed a derivative work subject to separate copyright protection. Bottom line--these are complex and surprisingly sophisticated legal issues and you need to retain IP counsel to advise you. The general advice you receive on this web-site is not a substitute.
The answer is different for each of the three sources, showing that it is not at all accurate to simply say "Nobody". The differences highlight the impact of technology and open source on copyright law.
First, Google Books: read the intro to en.wikipedia.org/wiki/Google_books where books retain their status. If public domain they remain public domain as Google has made those open source. If copyrighted, Google obtains a royalty agreement with the author or the book is not put on Google Books. Users simply pay a fee (either subscription or per copy) if the book is copyrighted, payment being made before downloading. Once you have the digital copy, you are in the same position as if you had bought a book. You can't make copies of a copyrighted book from your copy without committing copyright infringement, but you can freely copy if it is public domain. Google has lobbied for passage of legislated making compulsory licenses available for books so Google can realize its goal of digitizing every book ever written and adding the digital copies to its incredible database. If Google succeeds in this effort it will make every book ever written instantly available to every person on earth with Internet access, unless blocked by some Government censorship or the like.
Internet Archive (The Wayback Machine & archive.org) Read the intro at http://en.wikipedia.org/wiki/Archive.org As that intro states, "The Archive allows the public to both upload and download digital material to its data cluster, and provides unrestricted online access to that material at no cost." So, unlike Google Books, you never have to pay a fee to download content from archive.org. However, material downloaded retains its status just as with Google Books, so if a webpage has copyrighted material you should not use it for anything other than historical purposes, such a proving what a webpage looked like at any given point in time. For books on Internet Archive the situation is different, they are public domain since the Internet Archive does not digitize or accept copyrighted books, but only public domain books. So you can use books downloaded from the Internet Archive freely without restriction, unless there is a mistake due to someone illegally uploading a copyrighted book onto the Internet Archive database.
The Gutenberg Project (gutenberg.org) again, read the intro of the Wiki at http://en.wikipedia.org/wiki/Gutenberg_Project So, this one is strictly public domain, established back in 1971 for just that purpose. With only 38,000 books in comparison to Google Books with 10,000,000 already, this site is mostly a historical relic as the search engines at Google make Google Books much easier, and the same books are basically all online at Google Books.
The lesson to be learned: Lumping diverse sources of material together can result in faulty generalization. If you look specifically at your specific source, you will generally get your answer. If you don't know that source, chances are Wikipedia has an article to enlighten you.
So far, this is free to you. Until you pay a fee, I am not your lawyer and you are not my client, so you take any free advice at your sole risk. I am licensed in IL, MO, TX and am a Reg. Pat. Atty. so advice in any other jurisdiction is general advice and should be confirmed with an attorney licensed in that jurisdiction.