When we had the SARS outbreak in Toronto several years back we definitely stopped with the sharing of wine. We also stopped the sign of peace (shaking hands).
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In Roman Catholic churches as well. It’s been consecrated and I guess there’s no good way to store that. Extra hosts stay in the tabernacle. If a host is dropped, it’s picked up and eaten. If wine is spilled, it is wiped up and the cloth used is burned.
What’s the old joke? Wherever you find four priests, you’ll find a fifth! 
“Oh damn it, Seamus, we seem to have misjudged the wine again…”
In all seriousness, it’s usually the Eucharistic ministers (lay folks who assist in the distribution) who finish off the wine in the chalices. The presiding priest is one of the Eucharistic ministers (the “regular” one, in fact: All others need special (though very common) dispensation), but he generally handles one of the bread vessels, not a cup. And in most churches, the Eucharistic minister duty rotates amongst a fairly large group of volunteers, so it’s not even the same folks every week.
Hehe, this is amusing to me since, at least according to my dad, our family is descended from Levites (IIRC, he is, not sure if I qualify since my mom is Catholic and so am I)
Yep. Catgirls! ;j
What? 
Way back when I was still getting dragged to an Episcopalian church with my family, we took Communion from a common cup. We would go up to the alter and kneel next to each other, and they would come around and hold the cup up to our mouths so we could take a sip.
Well, one day, I was kneeling next to this guy, who, when the cup was offered to him, actually grabbed it and refused to let go. He was chugging the symbolic blood of the Christ like so much Mountain Dew…thirty seconds later, still slurping…the priest and I made eye contact and we couldn’t help but start giggling at the sheer absurdity of the situation…finally, the guy let go, and he’d finished off about two thirds.