Ellen,
I can see the birth of Anne Sweeney in Fintona on 16.7.1864 in the statutory records, on the irishgenealogy site. Father’s address at the time of the birth was the “United States”.
Statutory birth registration only started in Ireland in 1864 so if Mary was born before that you won’t get a birth certificate. You need to rely on church records. Fintona is in Donacavey RC parish. Their births and marriage records start in 1857.
I see a likely marriage for John Sweeney and Mary MacOsker in Donaghcavey on 26.10.1857. His parents were Alexander Sweeney and Mary McKenna. Difficult to make out the bride’s parents names.
I looked for Mary’s baptism 1857 to 1863 but did not see it in the church records on Ancestry. It’s possible that it’s there but not legible. Certainly many of the entries in that series of records are very hard to make out. You might want to search them yourself. The RC parish records are on-line free on the National Library site. You might want to go through them:
https://www.nli.ie/en/family-history-introduction.aspx