Native Swedish and Danish speaker here. I can confirm that Swedish people usually do not understand Danish, whereas Danes have an easier time with Swedish. I think Danish is just plain harder to understand, and I sometimes speculate that my early exposure to it has kind of hypertrophied my general language ability (I used to win awards for English vocabulary knowledge and that sort of thing in high school).

Interesting read on Danish infants and their slower rates of language acquisition: http://cphpost.dk/culture/quotdanskquot/danish-languages-irr...