I'll address only South American insurance, since I'm vaguely familiar with how it works. The subject has been discussed here many times, but generally discussions apply to one country--how to get insurance in Colombia, for example, or Peru--rather than the whole continent.
Loosely, and ignoring a few controversial bits and pieces, you'll need insurance in some countries but not others. When insurance is required, it's usually made available to you at or near border crossings--but not always. Insurance is specific to each country except in the southern tier, where you can buy Mercosur policies covering the whole block of countries.
In some countries no one seems to care whether you're insured or not, although this might change drastically if you are ever in an accident. In others, police and others appear to care only to the extent that they can extract bribes. In French Guiana, insurance was ridiculously expensive when I was there (and for a while it was apparently impossible, although green card insurance from the EU ought to be valid there), but most countries are more reasonable. Of course it's always worth reading the fine print on any policy you purchase, rather than relying on the assurances of whoever is selling the policy.
Just for fun, I typed "Colombia insurance" into the search box atop this page. I got a couple of pages of links to threads on the subject. I typed "Chile insurance" and got 10 pages of similar results. You might do the same, ignoring (unless you're really desperate for something to fill the idle hours) the very old posts. I'm not sure why you didn't find this useful, or whether you've tried it. Some past threads have give very precise directions, or at least GPS coordinates, to specific insurance sales agents.
Hope that's helpful.
Mark
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