Speaking from my experience of searching for Irish roots for my American immigrant born in early 1700s, it is difficult to find Irish BMD records, and parents and siblings, for persons from the 1700s.
My best luck finding some useful info was through the Belfast News-Letter, which started publishing in 1737. I searched for "John Shaw" from BNL inception up to 1800. I found a few mentions of that name.
24 Mar 1761, a notice was published saying that the Black Bull Inn in Lurgan (Co. Armagh), lately owned by John Shaw, now deceased, was now held by his widow Mary and son John Shaw.
26 June 1767, a notice said that this same inn was now solely owned by the son, John Shaw. There is no mention that his mother was deceased.
October 07, 1791: "The Black Bull Inn, Lurgan, to be let immediately, for such terms as can be agreed upon....Application for the house to be made to Mr. John Shaw, Belfast."
14 July 1789, a notice: "Ordinary at Lisburn -- John Shaw, at King's Arms, will have an Ordinary during the ensuing Races....Private Dinners as usual--good stabling, hay and oats, all in the most comfortable manner."
Lisburn is in Co. Antrim, across the Lagan River from Co. Down.
April 19, 1791: List of High Constables for the County of Down, appointed at Lent Assizes, 1791:
The list includes John Shaw of Drumaroad, for Kinalarty. I believe Kinalarty is a Barony.
These few mentions might give you some locations to direct your research. Evidently, these specific John Shaws remained in Ireland after the time when you think he emigrated to America. You did not say when John and Jeremiah Shaw emigrated, or if Jeremiah was born in Ireland or America. That might help others to reply.
I hope I don't err when I say I think you meant the 1790 census for South Carolina, which was the first federal census. If there was a 1700 census I'd welcome hearing about it. SC was a British colony then.
