There are languages like that---the distinction is referred to as "proximative" vs. "obviative". (Though strictly speaking, it differentiates between "more topical" and "less topical" third persons, which might not necessarily correspond with the order in which they're mentioned.) https://en.wikipedia.org/wiki/Obviative Apparently, there's even an Algonquian language which has a "further obviative" too, thus distinguishing three different levels of topicality.

Not Finnish, but in Turkish (another language without gendered pronouns) I'd use something like "o ne dedi bu ne dedi" (what did that say, what did this say)

I'd guess Finnish has more than one demonstrative pronoun, too :)

It's very informal and I'm not sure how widely spread it is outside of the Helsinki region, but at least least here in the Helsinki region, you can also use demonstrative pronouns (tää (= this), toi (= that)) as third person pronouns in certain specific circumstances to further specify how the people referred to in the conversation relate to you, the speaker, and whoever the listener happens to be. So you can have people A,B,C conversing, with D present but not participating in the conversation, and E not present but being discussed, and A can tell B "this told that that it/he/she did something" and it will be understood as "C told D that E did something". Not the exact distinction you were asking about, but it's another related axis of distinction in pronouns that I thought might be interesting enough to mention here.

I would use the idiom "sana sanaa vastaan", i.e. "word against word" for that situation.