Without a county of birth your task of identifying the place or places of birth of your ancestors in Ireland is difficult, but not necessarily impossible!
If you extend your name/location analysis work to Dixon’s, a much less common name in Ireland, you will discover:
1. Clusters of ‘Dixon’ in Dublin, which, like all capitals, attracts families towards it.
2. Particularly fruitful is the area contained by lines joining the towns of Kilcock, Trim, Athboy, Ballivor, Clonard, Edenderry, Carbury, and thence back to Kilcock. This 'hot zone' lies within the counties of S. Meath and N. Kildare.
3. Another cluster in Cork City.
4. A further cluster in North Wexford.
5. A small group in the Clare/Limerick area.
6. Likewise a small number of Dixon families in the Ballina/Belmullet area.
I discount Northern Ireland ‘Dickson/Dixon’ variants as obviously your ancestors were RC’s.
The names Bridget or Michael are too common in Ireland to assist, however name patterns in Irish families often repeat through the succeeding generations. So, if you were to consult the 1901 and 1911 Irish census, available free at http://www.census.nationalarchives.ie/
, you just might be able to identify a similar Christian name pattern in a particular area. Earlier census records were destroyed in the Irish Civil War. I imagine you should check ‘Duffy’ rather than ‘Dixon’ in that regard, as the ‘Male’ family usually predominates in ‘naming'.
Apart from those places you mention for Duffy there is a significant pocket of ‘Duffy’ in the East Galway area, as well as Galway and Mayo.
Have you checked port of entry and naturalization records, Ellis Island etc, which might easily assist your quest for identifying details?
The LDS site https://familysearch.org/
is particularly good at transcribing surviving RC Parish Records.
likewise provides a wealth of free research material.
And Good Hunting