Non native but have asked about this before (currently living in Norway south of Bergen since January 2014).

The school system is supposed to teach both but there is a lack of nynorsk in the media which causes more people who live in "nynorsk" areas to be proficient at both while those who live in primarily "bokmål" areas do not get as much exposure to nynorsk and find it annoying that it even exists. However, since both sides are supposed to learn the other at a young age they master both with ease. This holds true to the many dialects of Norwegian as well. To us non native speakers (usually passed age 12) the two written languages and many dialects all sound exactly the same. I still cannot hear the difference between Norwegian dialects, Swedish, and Danish but I easily notice it when it is on paper (I just assume I do not know what words they are saying cause I'm still new). However, to the native Norwegian they can hear the differences and the brain alters the words so that it is understandable.

Norwegian is actually 1 spoken language and not two. There are two different written forms of the language though. People tend to choose words more closely to how the people speak and write around them but the overall grammar stays the same. It is a bit silly because even the name of the country can be debated and both are present on passports (Norge and Noreg).

The two exist because of the Danish rule over Norway. After gaining their constitution the people began to try to move their language back to what Norwegian was before the Danish influence was added. This new form is called nynorsk while there are many who think it should stay how it was. That form is bokmål or "book tongue." It does not mean old Norwegian. That would be "gammelnorsk."

TLDR: nynorsk and bokmål are both written forms of the Norwegian language. Bokmål readers find Nynorsk noting because they do not receive enough of it when they are younger and find it annoying to even bother while nynorsk people find it annoying that they do receive so much bokmål in the media and schools. Most Norwegians do not seem to mind switching between the two when writing.

My actual complaint of all of this is that many are writing in their dialect which is even more ridiculous because that would make Norway have more than 6 written languages