The Irish Health system is a bit of a minefield when it comes to these type of tests. While it might seem straight forward, it could also present you with a few problems. I presume your GP has requested this test.
Lets say you get your private MR or Ultrasound and the result says you have a particular illness that requires hospital type treatment. What do you do next? You might be told you are now a private patient and you can only be treated by the private system.
Maybe your GP is happy to take the result and refer you to a public consultant, but you will still go on a waiting list to see that consultant.
I guess if the scan shows a serious illness or life threatening condition, you will be fast tracked, but that should be the case anyway, if your clinical indications are properly described by the referrer.