Private site to private site: YES. Customs in Ireland don't care when you're coming from the UK. No requirement whatsoever. The Gardai do not care either.
In NI, customs not required as it's still the UK, but yes definitely special branch.
All the paperwork etc. is for the UK side.
Things to watch out for in Ireland - flight plans are required for transiting all controlled airspace (and crossing he FIR boundary). This used to mean nothing unless you were transiting one of the regional airports, but recently the Dublin zone has expanded to include Weston, so now this is required for all flights into and out of there.
The Dublin area is quite busy, but other than that Ireland is wide open, uncluttered by airspace restricitons, and easygoing.
Flightplans are easily filed over the phone min 1 hour in advance, or 10 minutes before an airspace boundary.