There is no cross-border "hot pursuit" practice on the island, for readily understandable historical reasons. The PSNI's predecessor, the RUC, was particularly seen as an oppressive arm of the British state, and its presence south of the border would not have been welcomed. It was also often seen as being infiltrated by loyalist paramilitaries or sympathizers. For its part, the UK would not want to give the impression that it considered the entire island to still be its territory, subject to its law and institutions. Today there are also concerns in operational doctrine since PSNI officers are routinely armed and the Gardaí are not. (Hot pursuit is also not done from south to north, for essentially the same reason that not all communities in Northern Ireland would be happy about it.) There have been occasional incidents where the border has been crossed by accident, but I do not think any of those has led to court proceedings against the officers involved.
The issue is raised from time to time but the status quo remains: in the absence of a law giving PSNI officers authority to act in the Republic of Ireland, they can't do it.
- The default position for any two countries is that their police forces cannot act across the border. This is fundamental to sovereignty whereby each country has its own law.
- There is no universally recognized principle in international law allowing hot pursuit by police officers across a land border. If there were then it could be recognized at common law (in Ireland or the UK), assuming no statute or constitutional provision to the contrary. But there is no such principle, and several laws which point against it.
- Both jurisdictions have a notion of "citizen's arrest", deriving from common law but modified by statute. I do not know if that is available when the arrest is made by a constable acting in his official capacity, but outside his jurisdiction. It has not really been tested since the practice is not to pursue.
- Neither of these countries is party to Article 41 of the Schengen Agreement, which is the applicable hot pursuit law for the EU generally. There is a bilateral policing treaty, mentioned below, which deliberately does not include a hot-pursuit rule.
- UK law gives PSNI officers "the powers and privileges of a constable throughout Northern Ireland and the adjacent United Kingdom waters" (Police (Northern Ireland) Act 2000 s.32(2)), an expression which excludes the Republic of Ireland. This means that if a PSNI officer is anywhere else in the world – Singapore, Los Angeles, Dublin – then UK law does not consider them to be able to (for example) lawfully enter premises without a warrant in order to prevent an imminent breach of the peace, as they could in Belfast.
- For Ireland, the newly formed state legislated in 1923 for the formation of An Gárda Síochána in place of the Royal Irish Constabulary. There is no "legacy" statutory provision in Irish law by which a UK constable could claim automatic authority when south of the border. All such references were tidied up long ago.
- It is probably contrary to the Constitution of Ireland to allow a foreign police force to operate freely, outside the control of the institutions of the State. For example, Article 40(4) would seem to disallow someone to be deprived of their liberty and removed from the country in such a way that the Republic's courts could not intervene, as would happen if the PSNI snatched somebody in the south and took them back to the north. This is the same basis on which the Supreme Court struck down section 29 of the Petty Sessions (Ireland) Act 1851 in State (Quinn) v. Ryan  IR 70; that was a process allowing an English arrest warrant to be executed by the Gardaí directly, without a court process in Ireland. If that was disallowed, then letting the PSNI do it themselves is surely even less acceptable.
What does exist these days is a considerable degree of information-sharing, the possibility of prosecuting serious offences either side of the border, and various kinds of operational liaison. There has always been an element of informal contact but it is now officially supported to a greater extent.
Legally, there is a police co-operation treaty (made in 2002 as a follow-on from the Good Friday Agreement). This does allow officers of one force to move permanently to the other (Article 1), to be seconded to exercise police powers under the operational control of the other force (Article 2), or to visit without exercising police powers (Article 5). It is supplemented by a cross-border policing strategy, whose main focus is operational information sharing; instead of hot pursuit, one could radio ahead so that members of the other force can intercept on the other side of the border. But most operations are less rapidly paced – more about slowly gathering and sharing evidence than trying to handle a fleeing suspect. Note that a secondment under Article 2 has the effect of the PSNI officer effectively becoming a garda for the duration – this is different from exercising any power intrinsic to him as a UK constable.
The Sunningdale Agreement of 1973 previously led to the Criminal Jurisdiction Act 1975 (UK) and the Criminal Law (Jurisdiction) Act 1976 (Ireland). These enable prosecution of serious offences in either jurisdiction, extraterritorially. That is an alternative to extradition, which has also been politically fraught in the past. Extradition is presently available because of the Trade and Cooperation Agreement, confirmed in a recent CJEU case (C‑479/21 PPU), but previously worked under other law.