Sharing is called 分享 on the Taiwanese Facebook (and in my dictionary).

I think I've only seen 共 in compounds for things that are already shared, communal or co-existant, and notably as an abbreviation for communism. 中共 = 中國共產黨 = Chinese communist party. Not really a positive connotation. :)

(Disclaimer: Not a native speaker)

I don't know Kanji, but aparently in Japanese: "Affixing 共 to other people is disrespectful compared to 達 (たち, -tachi)."