At Dublin, arriving and departing passengers mix around the area of the 100 gates (formerly A pier) with signs directing arriving passengers through passport control and onto baggage reclaim. At all the other gates arriving and departing pax are segregated.
On the Schengen point, Ireland didn't sign up simply because the UK didn't. The thinking being that having to create a physical border on the island (between Ireland and Northern Ireland) was a) not acceptable and b) more bother than requiring citizens to carry a passport when travelling in the rest of the EU. It would also have removed the already existing free travel area between Ireland and the UK.
(with apologies to Adamm.'s earlier post which, now that I read it, already describes the situation at DUB
)
JAS