Also, how would the "typical peasant" be affected by suddenly having a different rule in terms of safety, freedom, economy etc?
There are several ways commoners may have learned of being under a new rulership. The most common was probably news and public announcements read by the priest after the sermon in church one Sunday, but written proclamations and propaganda was also used, as were town criers and similar people in other areas.
How they would be affected would differ by which "crown" they belonged to after their switch of rulers.
For example, Skåne, Halland, Blekinge and Bohuslän were annexed into the crown of Sweden and gained the rights and responsibilities of Swedish peasants. At some Danish estates in Skåne, the nobleman had the right to "neck and hand", which meant he could be the prosecutor, judge and executioner of his estate. This was very rare in Sweden and most if not all noblemen holding estates in provinces annexed to the Swedish crown lost that right.
The commoner might notice that civil servants and priests would be replaced with men loyal to the new order. In Skåne specifically the Swedish state attempted to replace priests who would not preach in Swedish and were loyal to the Danish regime. Tax collectors, judges, bailiffs, officers (of local garrisons or local raised regiments) and others would be replaced with new men - at times this benefitted the commoners, at times well-known and respected men were replaced by soldiers of fortune with little understanding or regard for local circumstances. The campaign to Swedify the areas annexed into the Crown of Sweden was heavy-handed and at times brutal and led to mutiple revolts and substantial guerilla warfare once the Danes attempted to retake those counties.
In other places, which did not become part of the Crown of Sweden chanegs might be slower, but still radical over time.
In Swedish Estonia, which remained its own Duchy with its own parliament and its own laws, Swedish authorities tried unsuccessfully to end serfdom but were unable to push it through the Lanttag, the Estonian parliament where the German burghers, priests and noblemen were represented, but not the Estonian commoners.
However, the Swedish authorities did end the right to "hand and neck" by the German nobility in Estonia and re-introduced the "move week" - a week yearly during which serfs had the right to move from one estate to another, should they so desire, which forced the German nobility to treat their sefs better. When the state seized substantial tracts of land during the reductions of Karl XI during the 1680s, serfs working royal land became tenants, and serfdom ended in all but name, since the nobility had to make the same reforms on their own estate or face all their serfs moving to royal land during the next "move week".
Swedish authorities in Estonia also eased language restrictions, allowing the Estonian commoners to communicate with the authorities in Estonian or German, and not just German as before. They also made an effort to translate the bible and Luther's catachesis into Estonian and Swedish priests who had learned Estonian travelled the countryside to teach Estonian commoners to read (but not to write, it was considered a separate thing back then) in order for them to study the bible and the catachesis.
An Estonian-Livonian university was also opened at Dorpat 1632 - while it mostly taught in latin, Estonian and Livonian were also used.