"They don't find out automatically, for now anyway.
But if you don't declare them and pay the tax due on time, then you are liable to fines and penalties for late payment in addition to the tax due if/when the Revenue come after you. The fines and penalties mount up quickly and can be three, four or more times the value of the tax due. They can go back indefinitely if the want to - the normal limitatioon of 3 years doesn't work if you are caught evading taxes."
I know that when you sign up for an etrade account you have to submit a signed tax exempt document (exempt from the US revenue) and that Ireland has an agreement with the US on taxation. This document I think contains your RSI number. However I am not sure whether that obliges etrade to inform irish revenue of profits or capital gains that you earn on stocks in your account. I think it merely allows them to give irish revenue information if they request it. However I am still in the dark on this issue. Also if this was the case, even if revenue performed an audit on you they still would not detect this account if you had no transactions between it and your irish accounts.