Wikipedia has this:

> ### Joint or separate possession
>For two nouns (or noun phrases) joined by and, there are several ways of expressing possession, including:
>
> 1. marking of the last noun (e.g. "Jack and Jill's children")
> 2. marking of both nouns (e.g. "Jack's and Jill's children").
>
> Some grammars make no distinction in meaning between the two forms. Some publishers' style guides, however, make a distinction, assigning the "segregatory" (or "distributive") meaning to the form "John's and Mary's" and the "combinatorial" (or "joint") meaning to the form "John and Mary's". A third alternative is a construction of the form "Jack's children and Jill's", which is always distributive, i.e. it designates the combined set of Jack's children and Jill's children.
>
> When a coordinate possessive construction has two personal pronouns, the normal possessive inflection is used, and there is no apostrophe
>  
> (e.g. "his and her children"). 
>
>
> The issue of the use of the apostrophe arises when the coordinate construction includes a noun (phrase) and a pronoun. In this case, the inflection of only the last item may sometimes be, at least marginally, acceptable 
>
> (e.g. ??"you and your spouse's bank account"). 
>
> [??/*"Tim and your bank account"]). 
> 
> The inflection of both is normally preferred (e.g. Jack's and your dogs), but there is a tendency to avoid this construction, too, in favour of a construction that does not use a coordinate possessive (e.g. by using "Jack's letters and yours"). Where a construction like "Jack's and your dogs" is used, the interpretation is usually "segregatory" (i.e. not joint possession).

("[General principles for the possessive apostrophe](https://en.wikipedia.org/wiki/Apostrophe#General_principles_for_the_possessive_apostrophe)", in "Apostrophe")

So in your example, unless they are writings that Giovanni and Walker co-wrote, you should use *Nikki Giovanni's and Alice Walker's writings*. Although I agree that it trips off the tongue better with just the second *'s*, and no doubt only the pedants in the audience would pick you up on it ;)