I have generally only heard people refer to professors as either "Dr. Smith" or "Prof. Smith". However, I received an email that referred to the keynote speakers at a conference as "Prof. Dr. Smith". Is this common? (It is for a conference in taking place in the EU and I am from the US if that helps).
German academia traditionally expects that one will use all relevant titles, so Prof. Dr. is pretty common there. Likewise other places with an academic system related to Germany in some way. As you note, in the US this would be very uncommon and the two titles you mention often used interchangeably there, even when it isn't clear that both apply.
And, if I remember correctly, it is always Prof. Dr. and never Dr. Prof. since the professorship was earned after the doctorate and is a "higher level" honorific. But "Herr Prof. Dr." doesn't fit that rule, I guess.
Background. In Italy, Dottore (short version dr. or dott.) refers to those who hold either a bachelor's degree, or a graduate degree, or a Ph.D., or those who are physicians.
Reply. Those who both serve as physicians and teach at medical school are commonly regarded as Prof. Dott. or Prof. Dr.
Yes, it is common to use both, and the reason is that these are two completely different categories. Doctorates are academic degrees that come in different flavors including "Dr. h.c." and "Dr. habil.". They indicate the academic level, just like Bachelor or Master degrees. For academic degrees, you typically use all degrees on the highest level, so you don't mention a Master degree when someone also holds a PhD, but you mention all doctorates. "Professor" on the other hand is (in almost all cases) an official title which you mention just like you would always refer to a judge as "Judge XXX", whether or not he/she's holds a PhD. That also explains the order: Titles come first, and you lose them when you lose the job. Degrees become part of the name (so to say, until recently, it was possible to have them on your ID card.)
All the other answers focus on the formal importance of titles and etiquette in German speaking countries. While refering to a person by his/her titles is normal, has not to be taken too far.
Still the level of formality isn't always the same. While the name and titles on a board can be meter long, normally an entitled person is approached by students as Herr/Frau followed by the most prestigious title and that suffices.
Professor will work well in your case. And this depends on situations, too. You might be for a beer after session and call American professors by name, it could be seen bizarre to switch to professor even, not to mention Mr or Sir professor :)
The language will often,if not always, be English, and somehow the level of formality goes along with.
I have worked in Austria and visited Germany. I never heard of herr Prof. Dr. or viceversa if not in pompous presentation as in special academic events. For sure students do not approach someone using two titles.
So there is nothing to worry about.
Edit: driven by other answers and comments I have focused on german speaking academia. I see the Q is more general. In a way the answer is still valid as Professor serves well the purpose of directly speaking to whom has that title (for foreigners, europeans or not, in Europe). Again printed material, official listing, ecc. might well detail all the titles. So Prof. Dr. might be encountered, but it will be always in the same fashion as the OP has already experienced, namely a third person or a third inanimate thing as a board introducing you an entitled person.
I have personally never seen Prof. Dr. X. At least from my experience Prof. is reserved for those who are teaching a course, but do not have a Ph.D. degree (graduate students teaching intro level undergrad courses) or faculty members who only have a MA/MS degree. Dr. in contrast is used for people who have a Ph.D. degree.