It depends on the statues. Mainly its DCC who has responsibility but some such as Wolfe Tone and Thomas Davis are under OPW control. The fountain on College Green for example is OPW. Others have been erected by interest groups in recent times, eg Phil Lynott.
It was common for most statues and memorials to be funded through public subscription (a very Victorian concept). You will see this on the dedication of statues such as Robert Emmet in St Stephen's Green.
Why none for Pearse? Good question. But very few 20th century figures are memorialised. It just fell out of fashion I imagine, or proved too controversial. None for any of the 1916 crowd except Markewicz (cant spell it). None for Dev or any post independence leaders or presidents. Michael Collins has a bust in Merrion Square, but I would suggest a small Michael Collins Square with the bust would be better.