No, whether you can directly apply for a residence permit from within the country is entirely up to each individual country and will not necessarily change after Colombian and Peruvian citizens are exempted from the Schengen visa requirement. There is no link between this and the Schengen regulations and no general rule that would apply in all Schengen countries.
In Germany and the Netherlands, some people can indeed directly apply for a residence permit based on their citizenship, even if they entered without visa with an eye towards residence. But the list of countries whose citizens can do that is much smaller than the list of countries whose citizens can visit the Schengen area without a visa. In both cases, it only covers a handful of high-income countries like the US, Japan, Canada, Australia or South Korea.
It turns out that the list of countries whose citizens benefit from this is almost the same in the Netherlands and Germany but I think that legally speaking it's mostly a coincidence. By contrast, in France, all third-country citizens must in principle first apply for a long-stay visa from outside the country and there is no exception based on citizenship.
There are always other exceptions as well, typically for refugees, sometimes also for people who qualify for a residence permit as of right (e.g. spouses) and in some other situations, so many details can make a difference.
So the most generic answer to your question is that you can't assume that anything will change or that it will become possible for Columbian citizens to switch from a short-stay to a long-stay status. But since you specifically mentioned Germany, I will go into a little more details about what I know about the rules there.
As explained in Switch from Schengen Visa to a Student Visa in Germany, citizens from 13 countries can in any case enter Germany without visa and apply for a student permit in the country. That list only includes one South American country (Brazil), even though most of them are already on the annex II list of country whose citizens do not need a visa to enter the Schengen area. Only citizens of Australia, Israel, Japan, Canada, South-Korea, New Zealand und the USA can do the same for a work permit (i.e. citizens of Brazil and other countries that belong to the first but not the second set of countries can do it for a permit that does not give them the right to work but not for a work permit).
Both of these lists are defined in § 41 of the Aufenthaltsverordnung. The same act also provides for a few other exceptions that can in principle apply to citizens of any country in § 39. Point 3 in particular implies that it's possible to apply for a residence permit within in Germany if the conditions required to obtain it only became fulfilled after entering the country (say you marry and therefore become eligible for a spouse permit or something like that). But I don't know exactly in which cases this rule can or cannot be applied.
Finally, the stay of members from the family of an EU citizen is regulated by an entirely separate piece of legislation and they can in any case apply for a residence card (technically that's not called a “permit”) from within the country.