You know, those vintage Art Nouveau liquor posters like these:
How do I find out who owns the rights to these? If I wanted to reprint them on umbrellas or mousepads or something, would I have to license the images from someone or are they in the public domain now?
If they are before 1923, they are in the public domain if you are a US resident.
The scans of them are in the public domain too, despite what a website may claim. Search on “Bridgeman v. Corel” on here to see me yell at people about this, on multiple past occasions. 
If they were copyrighted after Jan 1 1923, you will likely have to do a long and hard copyright search to determine if they are in copyright or not.
However, images on a website are owned by the website. You just have fair use rights only unless you scan your own.
A slight hijack but I have to ask, why is this exclusive to U.S. residents? I could understand the pictures being public domain for display in the U.S.A. but surely a person visiting America could display the images just as legally as long as they didn’t return to their home country with the images, where the copyright laws may differ.
You’re talking about two different things here. Fair use is intended to apply to works still under copyright. Works which are not under copyright are by definition not needing Fair Use as a defense.
A scan on a website which is a slavish copy of a public domain work is still a public domain work. Therefore it is not protected by copyright. If you mean that the public domain image as it exists on the server is “owned”, that is debateable only if you are talking about the original magnetic media. However, by the server operator allowing access to the file, they are allowing you to make a copy of said file, and they cannot control in any way, shape, or form what you do with that copy. The “image” is a file which rests on the web server; what you have on your PC is not that same image, it is a public domain copy of a public domain copy of a public domain work.
The only possible argument which can be made is that by agreeing to the ToS on a web site one may not be allowed to copy the image. However, there is absolutely no case law whatsoever on the subject of ToS’s restricting access to slavish copies of PD works on websites. In any event, there are no damages which can be assessed on copying of a PD work, there must be proof that the ToS was agreed to, and there must be proof that the file is the same file. Note that if the file has a digital watermark, then that could be construed as “creative content” added to the work, and therefore could be construed as being a derivative work. However, I am aware of no case law whatsoever which upholds that.
I’ve provided cites numerous times in the past on this subject. I’m sorry that people (in general) feel like they can slap their art book of old pictures down on a scanner and copy with impunity and with total protection, and yet restrict what happens with those slavish copies when they put them on the web, but they have no controlling legislation, case law, or precedent to support them. And for the most part, it comes down to the “mine mine mine!” defense on their part. I’ve fought with many site owners, including website operators of a couple of major museums, over this and I’ve always won.
mittu - I meant if you are working and operating your business within the US. If you are in a different country, copyright laws will vary tremendously.
Not true. Scanning an image does not give you special rights to that image. If the original art is public domain, scans of them are public domain as well (as long as you cut out/erase/cover up anything new that might have been added). Only actual added artistic vision can give something in the public domain a new copyright, and then that of course only applies to the new version and not the old. Any site claiming a copyright on a public domain image is basically either lying or confused about how the law works.
That goes for two dimensional images. Three dimensional images can have no copyrights by the act of photographing or scanning them, as the law so far recognizes that making something three-dimensional look good in two is an art form itself.