The only time I thought I might have a problem was doing exit immigration at FCO where I had entered the EU on my Irish, but at AC check-in had shown my Canadian passport which the agent then put the little sticker on. At exit immigration the officer flipped my passport over to look for the sticker (which wasn't there) but I had my Canadian passport also in hand and showed it to him and he nodded and waived me through.

As the laws (like in Canada) change that require you to enter your native countries on your native passports, the systems need to be upgraded to accommodate that. Interestingly my NEXUS profile has both on record and linked, occasionally at the US border I'm asked if I still hold my Irish passport, which I think is more about ensuring my information is up to date as they don't seem to care I have both.