As a vegetarian, and near-vegan who has been in this situation, I'd expect to be the one buying/maintaining veggie-exclusive equipment. That's what I've done. Where possible vegetarians/vegans try their best to live with others with the same diet. I lived in a vegan co-op for 2 years. It's for the best.

I am not anal about it, but if I know cookware has been used with meat, I generally won't use it. At restaurants or in others homes, I generally am able to suspend my meat-dar.

It has nothing to do with logic or the physics of food stickiness. If you've grown up that way, you have a visceral psychological aversion to mixed kitchens. It isn't religious either. It's the reaction meat-eaters have to the gross-out eating challenges on reality shows like Fear Factor, but more sensitive.

Many of my older relatives won't even eat in restaurants that maintain separate veg/non-veg sides of the sort Jonas mentions. They'll only eat food out of pure vegetarian kitchens.

People who go vegetarian/vegan later in life for consciously processed reasons tend not to have this deep aversion we born-vegetarian people do.