There are only two vectors for songs - people and media. The only media in the 1850s would have been paper, either printed or manuscript. So the question is how could people or goods have moved on land - or on water, sea, river or lake? As to your question about printing presses and methods - I can only say that, in Ireland, any small town, or even village could boast a printer, most of them using wooden 'common' presses, simple enough that a local carpenter might make one. However, a greater difficulty would be the availability of type - this was expensive - when new - but not so when old and degraded, which most of that used locally was.
The common ways for songs to get into papers or on to ballad sheets in Ireland, was by dictation, or copying - practices that led to the preservation of a lot of localisms or egregious errors.
If you're interested in the trade in Ireland search for John Moulden Printed Ballad in Ireland Aran Galway or paste https://aran.library.nuigalway.ie/handle/10379/5020