There is plainly more to say, but while awaiting specific responses to your query, it's worth noting that, in Ireland, it was far from the case that "literally half the country just upped and left". About 1 million people died in the great famine that afflicted the country between 1847 and 1851, which was of course also the main driver of emigration from Ireland in this period.

This is a big question, but a really interesting one. You haven’t specifically said so, but I’m assuming this is primarily a question about the nineteenth century, because that is the period of massive migration out of Europe to the Americas (and other places as well.) Quantitatively, you’re looking at about 50 million Europeans moving to the Americas between the 1840s and 1940s. Qualitatively, this is the source of many of the current residents of the Americas, especially in plains/agricultural areas of the U.S. and Canada. This migration is part of a larger out-migration from Europe to a broader range of destinations, but to keep this answer from getting too unwieldy let’s focus on Europe → America for now. (Belich, Replenishing the Earth, 2009) (It’s worth noting there’s also significant Asian migration to the west coast of the Americas over the same period.)

It’s true a lot of the historical literature on this migration focuses on “receiving countries” over “sending countries.” One practical reason for this is it’s easier to study presence than absence. That is, it's easier to locate and study immigrant enclaves in receiving countries than go back and locate the communities they left from. Compounding this difficulty is the fact that immigration records are often more complete than emigration records-- a lot of emigration statistics are reverse-engineered from immigration records or birth records. (Erikson, Invisible Immigrants, 1972 & Baines, Migration in a Mature Economy, 1985.) A more academic reason is that immigration to the Americas profoundly shaped post-independence nations: immigrants were the force behind settler colonialism and early urban growth and industry. Enough Europeans immigrated to the Americas to make themselves the majority group on a foreign continent, which is a huge historical transformation. Understandably, historians have tended to focus on that side of the movement.

All that being said, high rates of emigration seem like they must have had some effect on sending countries. However, effects varied because although European emigration is overall large, it is also extremely uneven. This is true on the country level, for example, there are a lot of British and Italian migrants than French migrants. But it's also true on a regional level. A big reason for this is that migration networks are largely personal: people are more likely to go to places they know people (aka “chain migration.”) For example, by 1950s, 3,000 residents of a Boston suburb had been born in the same small Italian mountain village, which itself only had a population of 4,000. This sort of migration might be meaningful for that particular village, but not necessarily as visible on the national level. In addition, migration statistics include different styles of migration. Some migrants are part of family groups or community groups that intend to be permanent settlers in their destinations. Others are temporary migrants, who intend to work for a few years and then return back to their home. Some are seasonal laborers, like Italian and Spanish agricultural workers who would travel to South American annually to work during harvest season and then return to Europe. (Adelman, Frontier Development, 1994)

What this all means is that migration tended to be concentrated in particular regions, industries, or life stages. The effects depended on how many people left, how long they left (forever versus a few years), and what type of person of left (entire families, young men, etc.) This suggests that the most extreme effects would likewise be regional, and therefore it can be hard to generalize. Broadly speaking, migration tends to be a symptom of a poor regional economy rather than a cause: people migrate because they can’t find work where they live. So emigration seems like it has less of an effect on sending countries, and more of an effect on receiving ones. What migration does sometimes do though is set up a flow of money back to Europe in the form of remittances. Historians know this happened, but I don’t know of any studies on their economic specific effects.

Politically, national responses to emigration vary. In the early- to mid- nineteenth century, while European population growth was high, emigration was usually not commented on, and sometimes even seen as a desirable way to prevent poverty and overcrowding. (Haines, Emigration and the Laboring Poor, 1997; Boucher, Empire’s Children, 2014). There’s also sometimes the attitude from sending countries that immigration will help “civilize” the new world-- this attitude is prominent especially in frontier regions. In the case of countries that have smaller territorial empires (Italy, Germany); migration is sometimes seen as a sort of proxy imperialism, a way to spread national culture. Towards the end of the 19th and beginning of the 20th century there started to be more concern about population loss, especially in Eastern Europe; this resulted in attempts to stop emigration through incentives to stay or barriers to exit. (Zahra, The Great Departure, 2017)

The cultural effects of widespread emigration on sending countries are probably the least studied effect. One important factor here, along with high rates of return, is the fact that many migrants did keep up communication with their country of origin. So emigration was not necessarily someone disappearing forever.

As someone has already noted, the Ireland example you’ve cited is a bit of an outlier because part of that population loss is from the famine; and emigration is in response to a specific catastrophe. This sort of migration is overall fairly unusual; most European migrants to the Americas were not fleeing disaster; and the population loss on the national level would be less drastic.