The question is fairly straight forward. "A Modest Proposal" was written by Jonathan Swift anonymously and published in 1729. I have heard at the time it became well known and there was some amount of outrage as some people didn't understand the satire. However, I cannot find anyone saying how it was published and got around so people would see it. It doesn't much seem like something I would expect the news at the time to print and I see sources saying he published it himself, but no more information.
A suggestion for further reading. A Modest Proposal was printed by Sarah Harding at Copper Alley in Temple Bar. On this subject James Woolley's Sarah Harding as Swift’s Printer, in Walking Naboth’s Vineyard: New Studies of Swift, University of Notre Dame Press (1995) would address the specific practicalities of the publication of that octavo pamphlet.
References: Mary Pollard. The Dublin Book Trade: 1550-1800. London Biographical Society, 2000.