I suppose for basic training it depends, for the Air Force given the non combat roles the RIAF will be doing should they absorb the cost of setting up light trainers for an around 100-150 pilots in total service. (so what a dozen a year). Maybe for helicopters pilots it would make sense for a full training system due to the larger number of units. Maybe when you get to modern simulators some in Ireland would be tied into the training systems in the UK.
For the Navy I can see doing it in Ireland (though Cork Harbour would have to be massively changed for the increased hulls and numbers), and then specialist training in the UK.
For the Army again basic training I could see but after that I think you'd have to use the larger training system of the UK. (the Glen of Imaal will take a pounding in this
).
I imagine that an attempt would be made to develop some level of Special forces, and with the decision to go to a Marine style force the SBS makes sense. (It could also be Ireland being unique as the other Commonwealth nations have SAS)