This is from Cnósach Focal again p157. Milleóigin: tá a milleóigín (=teanga) ar a cumas féin aici.

What does this fundamentally mean? She has got a sharp tongue and is always running others down?

Hard to say without more context, but accepting your translation of milleóigín as "language" (because I'm not familiar with the term) I would suspect something "the language she uses is up to herself" or "it's her own choice which language she speaks" with the idiomatic meaning being that she cannot be forced to use one language over another.

Well, "=teanga" was not my translation, but rather the translation given in the text of Cnósach Focal ó BB.

She is able (to use) her milleoigin?

What I would guess, as I haven't heard this would be something like along the lines of a similar meaning to tá a teanga ar a cómhairle féin aici, or that is what I thought upon reading it initially.

I've received an explanation of this from someone very knowledgeable: that milleóigín is a variant of billeóigín, little leaf, i.e. tongue, and the sentence means "she is very eloquent or witty".