Proper drunk, the kind where your eyes are all shiny and unfocused like you’ve been staring into the chippy fryer too long. It’s a pretty sweet, old-school way to say someone’s hammered without sounding nasty. Often used when someone’s cheerfully trolleyed rather than messy and aggressive.
"She was pie-eyed after three cocktails, swearing she could hear the jukebox flirting with her. Mate goes, Lightweight. She goes, Nah, they were doubles, now let’s get a kebab."