With some pairs of verbs, where one is subordinated to the other in some way, which one is negated may affect the meaning. Some languages might restrict which one can be negated, and leave the effect of the negation ambiguous, or resort to alternative organizations of the clause for the other meaning. Examples where which of the verbs is negated affects the meaning are easy to construct in English:
I didn't hear that he would arrive [...]
I heard that he wouldn't arrive [...]
Let's play around a bit with negation in general, and imagine consequences for constructions along those lines.
A reasonable idea for negation that exists in many natural languages and conlangs alike is to inflect the verb for negativity. Now we can imagine any number of restrictions here: only have negative forms for verbs that carry TAM marking (and have the subordinate verb be some type of infinitive). However, I'd like to up the ante beyond that: let's say only some verbs have morphological negativity marking (some even by root suppletion), and can be negated per se when infinitives, while other verbs take some negative particle, and cannot be negated when infinitives. Verbs that take particles come in several types, each which takes a different basic negation marker. The markers can also be replaced for some negative modalities, but the number of distinct negation markers in each modality beyond indicative is somewhat reduced.
Now, whenever a verb V2 is embedded in the VP of a verb V1, whichever of these two is negated gets its preferred negation if possible - if V2 is an infinitive and of the type that can be negated as infinitives, it is negated. If not, the negation migrates up one step (and creates some ambiguity), thus creating situations where
I don't know if he can do it
I know he can't do it
might not be distinct. However, alternative phrasings may provide ways out - i.e. changing one of the verbs, or rephrasing the embedded thing as a noun phrase or something.
Whenever both verbs have the same negation particle, the negation also becomes somewhat unclear – the marker might negate either one, or even both of the verbs.
However, indefinite pronouns (and some other determiners) or differential object marking may also help resolve the situation: only the direct arguments of a negated verb appear with negative indefinite determiners or with negative object marking on them (of course, the negative object marking might also be identical to the diff. obj. marking for imperfective verbs or somesuch, thus potentially reducing the usefulness of the d.o.m. for distinguishing polarity).
One could for that reason even have some kind of dummy pronoun, which is not an anaphor at all, but serves to either indicate that the last verb isn't negated, or that the first verb is negated - depending on whether it is like a negative indefinite pronoun or an "unnegative" indefinite pronoun.