Just because someone is a public figure does not mean that public figure has given up the publicity rights to his name and likeness. Public figure does not equal public domain. Many public figures trade handsomely off their names once they retire from public service (and sometimes, with a public figure like Arnold Schwartzenager, even before).
Arnold Schwartzenager sued over bobble-head dolls made with his likeness, for instance, as they infringed on his publicity rights. His attorneys said that his image is protected and valuable.
Most public figures won't sue, however. It makes for bad press. The same bobble-head doll maker who was sued by Schwartzenager also made Hillary Clinton and other famous public figure dolls for years with no problem.
So, it really depends on how litigious the public figure is (and possibly on whether he is running for election or re-election
).
Making parodies of public figures (on tee-shirts, or coffee mugs, or bumper stickers, or whatever), if they are
good parodies, are generally considered a fair use and not actionable (although nothing really prevents a lawsuit from being filed). Anything defamatory or obscene needs to be avoided.
As for marketing Sheriff Joe Arpaio merchandise, you can run your ideas by an attorney in your area, to see what sort of risk he/she thinks you are taking. And there is
always a risk when you are using someone else's name and likeness without permission and for profit.