Pretty much all, aside from a small number of city bus services are deemed as Public Service Obligation routes which are run under contract from the National Transport Authority (of which Transport for Ireland part of), to operators. The NTA supply the vehicles for the services and set the timetables and fares. The operators are allowed input and to give their views but the NTA makes the final decision. This has been the case for the last decade or so, when they were established. Before that things were directly operated by the Department of Transport who were rather more hands off.
In relation to Intercity services these are all commercial services where regulated competition is allowed which generally involves the NTA allowing two operators per license type per route on a first come first served basis on application for a route. Note that Express services and multi-stopping services are regarded as separate services, so for example you could have two Dublin to Galway non stop services and two Dublin to Galway multi-stop services but you couldn't have three of one of them.
In addition there is also a few clauses to stop predatory behaviour, which says that when there are two operators on a corridor, they must be time separated from each other 15-30 minutes at each town/city they call at, and ideally must operate from different bus stops. Operators cannot run extra services when they feel demand is high - if they want to add extra capacity, the extra capacity must be provided as a relief operating at the same time as the main coach. This was done based on the desire to avoid bus wars.
Where Bus Eireann has lost out is that it was slow to see the demand for high frequency non stop services and by the time that it realised the high demand, both of the Express licenses on the key corridors were already taken by private operators who saw potential demand and this has left Bus Eireann with most of the stopping services that they historically had whereas private operators have tempted those traveling between the bigger cities away from Bus Eireann by operating fast non stop services that run from very early to very late.
Cross border services between Northern Ireland and Ireland are effectively de-regulated as they are run under European rules so do not fall under the jurisdiction of either the Irish National Transport Authority or the Northern Ireland department of infrastructure, which is why we've seen what many people have described as a bus war on the Belfast - Dublin route with there being 3 express services in 15 minutes and then none for 45 minutes for most hours of the day. That kind of set-up would not be allowed on a domestic route in either country.
Licensing guidelines are here:
https://www.nationaltransport.ie/bus-licensing/download-licensing-legislation-guides-information/
A list of current licenses are here:
https://www.nationaltransport.ie/wp-content/uploads/2019/05/190510_CurrLic.pdf
