You can be a graduate of a foreign dental school and then come to the US to do your specialty training. Here at UB, we have lots of foreign dentists in our specialty programs because their home countries pay our state-school departments lots of money to train them in those programs. BUT, the expectation is that they will return to their home country to practice dentistry.
If you want to go to dental school in Ireland, then do orthodontics here and practice ortho here, well, you can't do it like that. In order to practice dentistry in the US, general or a specialty, you have to have an American DDS earned either through the traditional 4 year DDS curriculum, or the 2 - 3 year advanced-standing DDS curriculum available at some schools for international dental graduates. We have a woman in our dental class who trained as a dentist in the middle east, came to Buffalo to train in the Oral Pathology specialty and earned her certification as an Oral Pathologist. However, she has been in our dental class for the last two years because she had to enter the advanced-standing program to earn her American DDS to practice here. And this woman is not the exception. A woman in the class that just graduated did her dentistry in Syria, Endodontics specialty in Buffalo, and then had to do the 3 year advanced-standing program to get her American DDS to practice Endodontics in the US. I've talked to another resident in the Endo program from Syria who was considering going to the advanced training program at Pittsburg for another 2 - 3 years and an Oral Pathology resident from Mexico thinking the same thing.
So you would have to enter an advanced training program in the US to practice here after you did dental school in Ireland and orthodontics here. However, I believe this is a state specific issue, and there are a few states that will let you set up shop and practice without going through the advanced-training program. I remember reading a thread in the Int'l Dentist forum about the handful of states that will allow this. New York is NOT one of those states. There is also some exception about how you can work/teach in a US dental school with a foreign dental degree without going through the whole advanced-training program.
If you want to practice in the US, you should concentrate your efforts on getting into a US dental school because it will make practicing in the US much easier than if you are an international graduate. Many specialty programs hold spots for just for international candidates (like 1 out of 6 positions), but it is with the expectation that they will return to their home country after training.