The general law on death registration is that it has to be done by someone who was present at the death or, failing that, someone with a relevant connection to the deceased. So that could be a close relative or a neighbour, the Master of the Workhouse, a doctor at the hospital etc. That was the case in 1864 and is much the same today.
The requirement for death registration to be accompanied by a medical certificate or a coroner’s inquest was introduced around 1894. (I have a newspaper report from the Portadown News of 28th Oct 1893 on the work of the Select Committee which created that legislation, so presumably the requirement started after that work was completed).
I live in Ireland and I agree with eugenemcv that an obituary is the exception here. When there is one, they tend to be well known members of the community. The average man or woman doesn’t have an obituary, and never did.