I don't usually disagree with Susie (oh, the horror!), but in this case, I disagree. Yes, you can request a specific position, but you just have to do it in a way where you don't come off as a complete pervert. I've asked a good number of times when I'm seeing a new doctor for the first time. When I see a doctor that I've already seen in the past, it's a non-issue and I usually don't have to ask.
My preferred position is on the exam table on my elbows and knees, head and chest down on the table, ass up. I will always try to have it done in that position. But when push comes to shove (pun intended), I will comply and have it done in any position that they request. I can ask/request a position, but ultimately I go along with whatever they want. I've never had it done in stirrups -- that might be taking the request a bit too far.
If a doctor is about to perform a DRE on me, they would typically ask me to either stand up and bend over the side of the exam table, or to roll over onto my left size and pull my right leg up. Those seem to be the most popular ways to do it in the U.S. (in my experience).
If they want me to lay on my left side, I ask if we can do it some other way, since in the past I've twisted my back doing it that way (true). I've never had a doctor refuse to use another position when I say that. They certainly wouldn't want to risk messing up a patient's back. If they ask me to stand up and bend over the exam table, I'l politely ask if they can do it another way. Probably 90% of the time, they'll ask what I'd prefer. I tell them that in the past I've had it done on the table on my elbows and knees and that I've found it to be much more comfortable for me (true). I also add that I've been told by a previous doctor that it's much easier for them to reach my prostate in this position (also true).
At that point, the ball is in their court. They can agree and you can get into position. Or they can say no, and then you simply comply with what they requested in the first place. As I said, I've never asked to have it done in stirrups. That would be absolutely awesome to try, but in my way of thinking, that really may cross the line and give them the impression that I'm doing this just for my own kink.
The most important thing: Be polite, non-demanding and casual about the whole thing. And use some common sense and don't ask for something so outrageous that it is obviously just to satisfy your kink.