The BBC is actually broadcast in the Republic of Ireland (Eire) due to RTÉ paying the BBC to do so.
I don't believe this is true. You can receive some BBC channels on the Astra satellite or by cable, but the Irish broadcaster doesn't pay for it - they are competitors at some very small margin (e.g. showing EastEnders). The equivalent would be to pay a Scottish cable company for their subscription package including UK channels.
I had a look:
Historically, people in Ireland were able to get the BBC due to analogue spill over.
http://www.rte.ie/news/2010/0201/127070-communications/Also, "The Memorandum commits the two Governments [UK & Ireland] to facilitating the widespread availability of RTE services in Northern Ireland on a free-to-air basis, and BBC services in Ireland on a paid for basis."
It seems to be the Irish Government who pay for the BBC.
Also see:
UK Gov Website