To go on the record: constar en (el) acta.
To go off the record: sin constar en (el) acta, de forma extraoficial.

The context is obviously of legal nature, and so must be the translation too: Pure legalese. The standard correspondence of “on the record” in the Spanish legal jargon is “constar en acta”, and the opposite, “off the record”, is “sin constar en acta” or to say something “de forma extraoficial”. I do not see “off the record” anywhere in your excerpt, perhaps you have forgotten it or the second instance should be “then afterwards she says "OFF the record, now”. This is a bit confusing, but be that as it may, you have the two phrases available now.

Yes, there are several KudoZ entries that tackle this same issue or similar, but the only one that might be applicable to this specific context (a hearing in court) is the following (hope not to have overlooked any other…):
https://www.proz.com/kudoz/english-to-spanish/law-patents/14...


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Note added at 51 mins (2023-12-22 23:40:32 GMT)
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Nicholas, my immediate reply (still online). No, of course, we don´t say "antes de que constemos en acta", that´s wrong Spanish and doesn´t make any sense. What is for the record (to be taken, registered) is the content of the hearing, which is the subject of the sentence. Example:
"Antes de que la audiencia/el diálogo/el interrogatorio (whatever) CONSTE en acta (...)".
I hope to have made myself clear enough.

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Note added at 54 mins (2023-12-22 23:44:14 GMT)
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Another example:
Before going on record (...)
Antes de que [subject] conste en acta (...)
Or:
Antes de que [subject] se haga constar en acta (...)

In Spanish, a subject is required. Something must be the subject.

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Note added at 1 hr (2023-12-22 23:52:32 GMT)
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Alternative (by eliding the subject in the Spanish):
Before going on record (...)
Antes de que conste en acta (...)

However, the subject does exist, athough it isn´t expressed, it isn´t visible. The subject could be the hearing itself, a questioning, just a question, the statement of a witness, whatever! You can omit the subject in the Spanish counterpart, of course, but it does exist grammatically even if it isn´t expressed.