OK, to illustrate how this works, let's suppose your UK income from your UK employment is the equivalent of €50,000. (To keep things simple, we'll express all figures in euros and we'll just look at income tax, ignoring national insurance, etc.)
UK income tax on that will be the equivalent of about €7,000.
But, you're also liable to Irish tax on your UK income, against which you get a credit for the UK tax paid. Your Irish income tax liablity on a salary of €50k is about €7,850 . You get a credit for the €7,000 UK income tax you paid, leaving you with an Irish tax bill of €850.
Your total tax bill is (€7,000 paid in UK plus €850 paid in IRL =) €7,850.
Now, you start to pay contributions of €10,000/year to a UK SIPP. These are tax deductible in the UK, so your UK taxable income is now just €40k. UK tax on that is about €4,300. Great! You've saved €2,700 in UK tax!
But your Irish taxable income from your UK employment is still €50,000 — I don't think you'll get a deduction in Ireland for contributions paid to a UK pension scheme. Irish income tax on €50k is still €7,850. You get a credit for the UK tax you paid, so your Irish tax bill is now (€7,850 - €4,300 =) €3,550.
The total of Irish and UK tax that you pay is now (€4,300 paid in UK plus €3,550 paid in IRL =) €7,850. In other words, you haven't saved any tax at all. You pay less tax in the UK and more tax in IRL and it all comes out in the was