Shouting someone a drink in the pub is an expression that originates in Australian and New Zealand English, with the first citation dating back to 1854 in what is clearly a goldfields context. One theory is that the person who is shouting is the one who yells the order to the bar, but there is an interesting citation in which a newcomer has the business of shouting explained to him in the sense of everyone taking a turn to buy drinks, and, thinking he understands, he yells the request to the bar. The barman and the other drinkers are all very amused. This would seem to imply that the obvious understanding of shout is not correct. The word is put in quotes to make it clear that it is not your usual shout that is referenced here. So what could it be? There is a Scottish English word shot meaning ‘payment of a bill, reckoning’ used particularly in tavern accounts. There is an illuminating citation in the English Dialect Dictionary:
Now, chaps, whad’n’ee a to drink -- ale or short [spirits]? – an’ I’ll stand shot.
This seems to be exactly the context of the Australian shout.
The theorised Irish origin is in an Irish Gaelic phrase glaochardheoch, ‘to call or shout for a drink’. This can only have been used by one Irishman to another Irishman. No one else would understand it.