Yes they can find out, for your information they even know that you are economically active in UK and Ireland, also if you have applied for a UK residence and got refused and applying for residence in Ireland than they will easily find out when you will go to the the immigration counter for visa extension or first time registration. All the data has been shared.
Source:
· Immigration data sharing with the UK. In 2014, Minister Fitzgerald and the UK Home Secretary signed a new agreement for the sharing of immigration data between both countries’ immigration authorities. This comprehensive agreement is aimed at protecting the Common Travel Area (CTA) from abuse as well as providing the means on which to extend the benefits of borderless travel between the two countries to genuine visitors through initiatives such as the British Irish Visa Scheme. In 2014, under data sharing arrangements the details of over 100,000 Irish visa applicants were checked against UK records.
http://www.inis.gov.ie/en/INIS/Pages/Im ... %93%202014
