Who owns the right to reproduce your painting?

You, the artist, do. If a buyer copies it without your consent, he’s liable for damages.

The Monarch of the Glen, 1851, Sir Edwin Landseer, has been used for everything from the Hartford Insurance Company’s stag logo to biscuit tins and butter wrappers. You can read its history here
“A previous customer wants to use the painting they purchased as part of an advertising image. Anyone with experience in this?” an artist asked on Facebook. It’s a confusing subject.
The fact that her client approached her first indicates he knows that royalties are due. The artist needs to research royalties in her discipline before she can negotiate. I don’t envy her; it’s a difficult thing to pin down. 
Copyright law in the United States is different than in other countries. We have some treaty protection for our work, but nothing really protects artists from international freebooters. What I’m about to say is applicable only here. And of course I’m not a lawyer, so my advice is worth exactly what you’re paying for it.
You don’t need to register your copyright with the government for it to be protected. Copyright exists naturally once the work of art comes into existence. For something to be copyrighted, however, you actually have to make it in a tangible form. “I had that idea first!” is not a valid copyright argument, unless you can demonstrate that you actually acted on the idea.
Copyright gives you, the artist, the sole right to reproduce, create derivative works from, make prints from, and display the work publicly. It doesn’t matter if you sell the physical painting or sculpture; the rights to its image remain with you.
The Associated Press sued artist Shepard Fairey for copyright infringement for his Obama poster. The parties settled out of court in 2011, with details remaining confidential.
‘Work for hire’ is an exception to the general copyright rule. It happens when an employee creates art as part of his job or is specifically commissioned as part of a collaborative work. Advertising art done by either a salaried artist or a contracting artist is an example of the former. An example of the latter might be the artists who draw the illustrations in a medical dictionary for a commercial publisher.
An agreement that the work is for hire isn’t sufficient; the project must meet the courts’ narrow definition. Whether or not the artist is attributed makes absolutely no difference to its legal status.
String of Puppies, 1988. Jeff Koons.
Another exception is the fair-use exemption. This permits limited use of copyrighted material without the copyright holder’s permission. Fair use exemptions include research, scholarship, commentary, criticism, news reporting, parody, and search engines. Without it, artistic commentary (including mine) would grind to a halt. 
The fair use exemption doesn’t allow unlimited copying of artwork by other artists, even for self-styled ‘appropriation artists’ like Jeff Koons.
Puppies, 1985, Art Rogers. Rogers successfully sued Jeff Koons for copyright infringement, one of several times the artist has been sued.
Copyright currently lasts through the life of the author plus 70 years. If the work was produced under corporate authorship it may last 120 years after creation or 95 years after publication, whichever comes first.
What should you do if you find your painting gracing a wine label or a set of plastic dishes—or worse, badly reproduced? Consult a lawyer specializing in intellectual property. That’s theft.

Of course, anyone involved in an intellectual exchange should have a lawyer. The law is extremely complex, and expert advice is worth every penny you pay for it.