"I leaked nothing to nobody" is not considered standard English, as far as I can tell. It is possible to use multiple negative words in a single sentence wtihout them cancelling out in some circumstances, but I don't think this is one of them. (A random example I can think of that would be standard is "I wouldn't sell the farm, not to him.")
- Non-Standard Dialect: He didn't give nothing to nobody.
- Standard: He didn't give anything to anybody.
(I-Language: An Introduction to Linguistics as Cognitive Science, by Daniela Isac & Charles Reiss)
"I leaked nothing to anybody" is perhaps "technically" correct, but it sounds unusual to me; certainly less natural than "I didn't leak anything to anybody."
It often sounds more natural for negation to be marked "at the sentence level" (by a negative auxiliary like "didn't," or an auxiliary accompanied by the negative particle "not") than with another specific negative word. This applies equally as far as I know to dialects with negative concord: the Yale Grammatical Diversity project overview of this phenomenon only covers cases where specialized negative words occur alongside sentential negation, not in place of it. It is only a short summary, however, and it does point to a further source you may wish to consult:
There is variation in the types of negative concord that different English varieties allow. For an overview, see Smith (2001).
Clause-level or verbal negation is also the most usual/"expected" state of affairs from a typological standpoint ("On the typology of negative concord," Johan van der Auwera & Lauren Van Alsenoy); that is, sentences structured like "John bought nothing" (taken from van der Auwera and Van Alsenoy) are uncommon worldwide. So I find sentences like the one you quote here, where clause-level negation is not used but multiple other negative words are used, to be interesting and somewhat surprising.
I did find other examples of this type of negative concord, if that's what it is (multiple negative words but no clause-level marker of negation), on Google Books:
You can see that the examples seem to come from a diverse group of speakers. Negative concord is a feature of AAVE, but it is not at all a distinctive feature; it's widespread in all sorts of varieties of English, particularly colloquial speech.
I don't find it at all plausible that the statement was intended as "weasel wording" or "litotes."
Note that people speaking spontaneously often don't produce grammatically correct, complete English sentences. False starts, self-interruptions and tangents are the norm. I find Andrew Brēza's suggestion ("Listening to the audio recording of the statement makes it sound like she was abbreviating these two thoughts into one sentence: "I leaked nothing. I leaked to nobody." If she were writing her remarks as part of a speech I'm sure she would have worded it differently.") very plausible.
Edit: I just noticed there is a Language Log post on the topic by Mark Liberman. He refers to it as "emphatic multiple negation." I would recommmend reading it and the comments below.
Unfortunately, I don't have access at the moment to a grammar that explains whether this exact use of negative words with each other is standard or non-standard. (The Isac and Reiss quote at the top of my post uses an example with a negative auxiliary "didn't".) However, here are some Google Books snippets that I found that I think back up my feeling that sentences with two specific negative words ("n-words") and no clausal negation like "Nobody did nothing" are cases of negative concord and not generally considered standard:
As Alison Henry points out, they are Positive Polarity Items in the strong sense that even in Negative Concord dialects, which allow, for instance Nobody did nothing as synonymous with Nobody did anything, the examples in (5) remain ... (Syntax at Santa Cruz, Volumes 1-3)
Juan never to nobody did nothing. Juan never did anything to anybody. The traditional grammars claim that the Negative words before and after the verb have to be different. Because this seems too obvious, it is better to claim that... (Chicago Linguistic Society. Regional Meeting - 1991)
For example: Nobody did nothing means everybody did something. The correct negative sentence is NObOdy did anything. (Test Your Grammar, Rachel Bladon)