Somehow I don't like these expressions. For me, they mean approximately "to do some favor to public" as opposed to "to do smth for myself".
But church is just small subset of public. Hence, it seems that there's no simple way to say that in short words. "На общественных началах" is probably best, IMHO.
Context and intonation plays role. "Заняться кем-л." means approximately "to pay more attention to somebody". It could be about starting to flirt ("я ей займусь"), or about something else.Я им займусь
When I had seen your post, I recalled old movie Association de malfaiteurs (1987) - IMDb
"Козёл! Козёл до кончиков ногтей! Я им займусь" - says "good guy" about "bad" Hassler in Russian translation. Here "заняться Хасслером" means "to punish", "to bring to account", ... something like that.
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