This is all getting very contrdictory and confusing,

The replies given by Gagravarr are largely incorrect, please disregard them.

As has been mentioned the UK and Ireland (and the Channel Islands, and the Isle of Man) form what is known as the Common Travel Area or CTA.

There are normally no passport checks between the different parts of the CTA, including the Republic of Ireland which unlike the Channel Islands and the Isle of Man is a sovereign state. However, there is one exception to this (which has existed for a few years) - anyone flying into DUB (only) from outside the Republic now has to go through a checkpoint. British and Irish citizens can get through the checkpoint using their driving licences or other acceptable form of identity, others need to present a travel document - passport or EU/EEA National ID Card. There are no immigration checks travelling in the other direction, nor flying into other Irish airports or going by sea. It's just a Dublin thing.

Flying to Ireland via the UK you go through Immigration in the UK. You don't just go through a security check as previously suggested. You fully enter the UK and then continue on an internal flight.

I don't believe either the UK or Ireland have a requirement for dual nationals to use the "local" passport - but it makes sense to do so as EU citizens enter EU countries as a matter of right, not privilege.