An Irish citizen living in the UK has the same rights as a UK citizen. A UK Citizen living in Ireland has the same rights as an Irish Citizen, with the exception of not being eligible to vote in the Irish Presidential election. There is no immigration control between the two countries.
Yet!
I think that you may find that many more "Irish citizens" exercise their right to vote in the UK than UK citizens do to vote in Irish matters. Perhaps they feel more strongly about adversely affecting us than we do them.
All citizens of Commonwealth countries have the right to vote in UK elections. (So, for example, Canadians, Australians, Nigerians and Bangladeshis can all vote here).
Although Ireland left the Commonwealth in 1949, their citizens retained their voting rights through the provisions of the 1922 Anglo-Irish Treaty, which was signed to bring the Irish Free State into existence. (The rights were retained when the Irish Republic was created in 1937)..