There is so little off road riding in Ireland you'll need to plan a road trip. All land is privately owned, there will be minimal to no wild camping opportunities, there are also very few campsites.
I'm going to call BS on that. There is loads of riding in the country, not all land is privately owned, there are lots of campsites, you can wild camp sundown to sun up like any other European country.
There are big differences to the US with regards to camping in designated sites in forests certainly - there is no official right to camp like Scotland certainly - but in over 35 years of walking,riding, running in Ireland I've never had issues putting up a tent/bivi to sleep once outside of national parks.
OP - lots of riding opportunities once you're willing to sit down and look at a map. Officially there is no designated difference between trails/footpaths/bridleways in Ireland. Once you're not randomly tramping across a farmers land, but on a designated right of way, you're golden. Walkers may believe they have the only access rights to trails, they are however misinformed.