You don't have to mortgage it locally. You can - or you can borrow against an existing property/asset here - or a combination of both. The differences will centre around:
1) interest rates ... and all ther terms and conditions of each mortgage offer. Generally, the Irish one will be cheaper, lower interest rate and be a lot easier to qualify for. The foreign mortgage has the advantage of being of 'no recourse' to your property here; the security the bank has is over the deeds of the foreign property, not the Irish one.
2) your desire to tie up equity in Ireland if you choose to mortgage it here.
Speaking very generally, if you have no lump sum or cash to use as equity, what many people do is to borrow enough from their irish bank and limit their foreign bank borrowing to that monthly amount that is covered by the rent; in other words, borrow enough in Ireland to ensure that you have surplus cash each month in your foreign bank account.