In this event you would have two occupational pensions from the public service. The only potential issue is "pension abatement", which means that the combined amount from your salary on re-employment plus the old pension payment cannot exceed the uprated salary from the post from which you originally retired. As your pensionable age in the existing scheme is (presumably) 65 this would not be an issue prior to that - and then only a potential issue.
Your state pension would be based on your total PRSI record in the usual way, ie, all your full rate contributions (and credits), whether these came under public or private sector employment.